Capacitary Riesz-Herz and Wiener-Stein estimates
|
|
- Hollie Clarissa Richards
- 6 years ago
- Views:
Transcription
1 Capacitary Riesz-Herz and Wiener-Stein estimates Joint work with Irina Asekritova and Natan Krugljak J. Cerda, Departament de Matema tica Aplicada i Ana lisi; GARF, Barcelona
2 Riesz-Herz equivalence for maximal functions Maximal functions on R n, used to control many operators: Hardy-Littlewood maximal function Average function Mf(x) := sup Q x f (t) := t t 0 f(x) dx. Q Q f (s) ds, F. Riesz (932) proved the pointwise estimate (Mf) (t) f (t) f f and Herz (968) the reversed estimate. So the Riesz-Herz equivalence is f (t) (Mf) (t) Recall tf (t) = K(t, f; L, L ) = inf { f 0 + t f } f=f 0 +f and the Riesz-Herz equivalence may be written K(t, f; L, L ) t(mf) (t).
3 Wiener-Stein equivalence Riesz-Herz can be proved starting from the Wiener-Stein estimates: { A nt {Mf > 2t} f(x) dx 2 n t Mf > t }, c n { f >t} Wiener (939) the first to prove an ergodic theorem. Stein (969) reversed the estimate (using Calderón-Zygmund) to prove f L log L if R k f L. Since and f(x) dx ( f(x) t) dx f(x) dx 2 { f >2t} { f >t} { f >t} { f >t} ( f(x) t) dx = dist L (f, B L (t)) = E(t, f; L, L ), the Wiener-Stein estimates can also be written as A n 2 t {Mf > 4t} E(t, f; L, L ) 2 n t {Mf > t/c n}, E(t, f; L, L ) t {Mf > t} for short.
4 The functionals K(t, f; A 0, A ) and E(t, f; A 0, A ) K(t, f) = inf { f 0 A0 + t f A } f=f 0 +f E(t, f) = d A (f, B A0 (t)) = inf{ f tg A ; g A0 }. K(s, f) t E(t, f) = sup s>0 s Fact: E(t, f) < f A 0, closure in A 0 + A. f p (A 0,A ) θ,p := 0 (t θ K(t, f)) p dt t 0 (t α E(t, f)) pθ dt t (θ = /(α+))
5 Fractional maximal functions Related to Riesz potentials: M αf(x) := sup Q x Q α/n Q f(x) dx (0 α < n), with Riesz type estimate (Cianchi, Kerman, Opic and Pick) (M αf) (t) c sup t<τ< τ α/n M α bounded on certain Lorentz spaces. τ 0 f (s) ds Sharp: holds for the spherical rearrangement f (x) = f (ω n x ). But this estimate cannot be reversed.
6 New maximal functions Edmunds and Opic (2002, potentials with logarithmic smoothness): M h f(x) := sup f(x) dx Q x h( Q ) if h(t) = t λ [( log t) A χ (0,] (t) + ( + log t) B χ (, ) (t)] (0 < λ ). Q Berezhnoi (999), M h with h measure function (continuous, increasing, 0 only at 0) satisfying the condition 0 h(t) γ i h(t i) if t i t i. E.g. h quasi-concave, or as in Edmunds-Opic when 0 A, B λ. Maximal capacitary functions: M Cf(x) = sup f(x) dx. Q x C(Q) Q
7 General capacities A capacity on R n will be C : B [0, ] on Borel sets s.t.: (a) C( ) = 0, (b) increasing, and (c) subadditive Our capacities will be Fatou: A k A C(A k ) C(A). A corresponding outer capacity C is defined as { } C (A) := inf C(Q i); A Q i i= i= ( C(A)). Analysis based on Choquet s integral: If f 0, f dc := 0 C{f > t} dt. f L p (C) := ( f p dc ) /p = f C p, f C(t) = inf { λ > 0; C{ f > λ} t }. Examples: Cap E (K) := inf{ u E; u Lip 0, u = arround K, 0 u }. If E is a function norm, A E := χ A E. Classical Newtonian capacity of a conductor. Hausdorff contents.
8 Main example: Hausdorff contents h a measure function (continuous, increasing and 0 only at 0) s.t. lim t h(t) = and Λ h (Q) h( Q ), Λ h (2Q) 2 n Λ h (Q), Λ h(a) = Λ h (A), and h(t) γ h(t i) if t i= i= i= { Λ h (A) := inf h( Q i ); A Q i }. Λ h (Q k ) as Q k. If C (A) γc(a) (i.e. C (A) C(A)), we call C of Hausdorff type. t i i=
9 Newtonian capacity K R 3 a conductor, µ a charge distribution on K. (Coulomb s law) Electrostatic field created by µ at x R 3 : x y dµ(y) E(x) = dµ(y) = x y 3 x y = U µ (x). Gauss: the charges move to equilibrium µ e and U µe K = Vµ, constant. Newtonian capacity C 2(K) := µ(k) V µ = inf independent from µ. C 2 is extended to an outer measure by defining { 4π ϕ 2 2; ϕ Lip c (R 3 ), ϕ arround K C 2(A) = sup C 2(K) (A R 3 ). K A Borel sets are not measurable (lack of additivity) but they are regular: C 2(A) = inf{c 2(G); A G}. C = Λ h with h(t) t, since the capacity of a ball is the radius. If C is any Riesz capacity or a Sobolev p-capacity, then C Λ t α. },
10 Our results Riesz-Herz type estimates: K(t, f; L, L,C ) t(m Cf) C. Wiener-Stein type estimates: E(t, f; L,C, L ) C{M Cf > t} for short. Morrey space L,C = L,C (R n ), with f L,C := M Cf <, is a substitute for L (R n ) ( f = Mf ). We will suppose that C satifies: (A) Doubling condition: C(2Q) γc(q) for any cube Q. (B) Q i R n lim i C(Q i) =, and (C) C C (C is of Hausdorff type ).
11 Dyadic setting and D = k= { n } D k = [(2m i )3 k, (2m i + )3 k ]; m i Z, (i =,..., n) i= Successive ancestors of D : D D 2 D j R n MCf(x) d := sup f(x) dx, D x C(D) f L,C d D k D := M d Cf <. E d (t, f) := dist L (f, B,C L (t)) (f L + L,C, t > 0). d { Cd(A) := inf C(D i); A i= i= } D i, {D i} D.
12
13 Basic equivalences M d Cf and M Cf are not equivalent, but, using that Q there exists D Q D k s.t. D Q Q D Q, and D Q is the union of n cubes D i D k : (a) C (A) C d(a) n γ 4 C (A). (b) L,C = L,C d, with equivalent norms. C(Q) C(D Q ) γ C(D Q) 4 γ C(Di) 4 n f(x) dx γ 4 f(x) dx n γ 4 sup f(x) dx C(Q) Q C(D i= i) D i D C(D) D (c) C{M d Cf > t} C{M Cf > t} γ 2C{M d Cf > t/γ }.
14 C{M C f > t} γ 2 C d {M d C f > t/γ }, γ = n γ 4 Maximal non-overlapping with maximal cubes s.t. { MC d > t } D, C( γ D) { Cd MC d > t } + ε γ Take D Q Q D Q, D i0 D Q, D D x {M Cf > t} f(x) dx > t (x Q) C(Q) Q C(D i0 ) C(Q) Q f(x) dx γ4 n i= D i0 f(x) dx n γ 4 C(Q) C(D i ) Q D i f(x) dx. f(x) dx > t γ. { M d C > t γ } D i0 D j, x Q D Q 2 D j. {M C f > t} j 2 D j C{M C f > t} γ 5 j C( D j ) γ 5 (C d { ) MC }+ d f > t ε. γ
15 Wiener-Stein type estimates E(t, f) := dist L (f, B L,C (t)) = f in L, or in the closure of L in L + L,C, i.e., Then C(D) D f(x) dx > t limj ancestor D j whith Theorem C(D j ) inf f h, h L,C t E(t, f) < for all t > 0. f(x) dx = 0 D has a largest C(D j ) D j f(x) dx > t. D j E(t, f) tc{m Cf > t}. For the proof we move to the dyadic setting and show that tc{m d Cf > 2t} E d (t, f) tc d{m d Cf > t}.
16 Proof of E d (t, f) tc d {M d C f > t} {MCf d > t} D i E d (t, f) t C(D i)(< )? i= If neighbors of D i0 in D i 0 all in {D i}, substitute by D i 0. Necessarily D N+ i {M d Cf t} = for some N. i= A neighbor of D i in Di meets {MCf d t} and 5D i Di : f(x) dx t. C(5D i) 5D i Since fχ {M d C f t} L,C d t, E d (t, f) f fχ {M d C f t} = fχ {M C d f>t} f t C(5D i) γ 3 t C(D i). 5D i i i i So E d (t, f) γ 3 tc d{m d Cf > t}, since C d(a) = inf{ i C(Di)}.
17 An application to interpolation Characterization of (L, L,C ) /p,p ( < p < ) as the space of all f L loc(r n ) such that M Cf L p (C) < : M C : (L, L,C ) /p,p L p (C), f (L,L,C ) /p,p M Cf L p (C). For the proof (cf. [BL]) we have that, if ϑ = /p = /(α + ), f p (L,C,L ) /p,p := From Wiener type estimate: M Cf p L p (C) = p 0 0 For the inverse, use Stein type estimate. 0 0 (t ϑ K(t, f; L,C, L )) p dt t (t α E(t, f; L,C, L ) ϑp dt t. t p C{M Cf > t} dt t p E(t, f; L,C, L )) dt t = f p (L,C,L ) /p,p.
18 Riesz-Herz type estimates Theorem For every f in the closure of L in L + L,C, K(t, f; L, L,C ) t(m Cf) C (t > 0). Wiener-Stein used as follows: K(t, f) inf f g + t g L,C E(s, f) + st. g L,C s By Stein type estimate: E(s, f) c 3sC{M Cf > c 4s}. If we choose c 4s = (M Cf) C(t) + ε, then C{M Cf > c 4s} t K(t, f) E(s, f) + ts c 3sC{M Cf > c 4s} + st c 3st + st and K(t, f) c 4 (c3 + )t((mcf) C(t) + ε). So K(t, f) t(m Cf) C(t). The reverse estimate is proved using Wiener type estimate.
19 L p -Morrey norms ( p < ) Associate Morrey norm: ( ) /p M C,pf(x) = sup f(x) p dx Q x C(Q) Q f L p,c = ( /p. sup f(x) dx) p Q C(Q) Q Herz-Stein type inequalities for these Morrey norms: t /p (M C,pf) C(t) K(t /p, f; L p, L p,c ) (f L p ). Proved using the power theorem of interpolation theory. ((M C,pf) C(t)) p = (M C f p ) C(t) t K(t, f p ; L, L,C ) t K(t /p, f; L p, L p,c ) p.
Herz (cf. [H], and also [BS]) proved that the reverse inequality is also true, that is,
REARRANGEMENT OF HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN LORENTZ SPACES. Jesús Bastero*, Mario Milman and Francisco J. Ruiz** Abstract. For the classical Hardy-Littlewood maximal function M f, a well known
More informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationJUHA KINNUNEN. Harmonic Analysis
JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes
More informationBorderline variants of the Muckenhoupt-Wheeden inequality
Borderline variants of the Muckenhoupt-Wheeden inequality Carlos Domingo-Salazar Universitat de Barcelona joint work with M Lacey and G Rey 3CJI Murcia, Spain September 10, 2015 The Hardy-Littlewood maximal
More informationOn pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa)
On pointwise estimates for maximal and singular integral operators by A.K. LERNER (Odessa) Abstract. We prove two pointwise estimates relating some classical maximal and singular integral operators. In
More informationREARRANGEMENT OF HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN LORENTZ SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 1, Pages 65 74 S 0002-9939(99)05128-X Article electronically published on June 30, 1999 REARRANGEMENT OF HARDY-LITTLEWOOD MAXIMAL FUNCTIONS
More informationON A MAXIMAL OPERATOR IN REARRANGEMENT INVARIANT BANACH FUNCTION SPACES ON METRIC SPACES
Vasile Alecsandri University of Bacău Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics Vol. 27207), No., 49-60 ON A MAXIMAL OPRATOR IN RARRANGMNT INVARIANT BANACH
More informationMATH6081A Homework 8. In addition, when 1 < p 2 the above inequality can be refined using Lorentz spaces: f
MATH68A Homework 8. Prove the Hausdorff-Young inequality, namely f f L L p p for all f L p (R n and all p 2. In addition, when < p 2 the above inequality can be refined using Lorentz spaces: f L p,p f
More informationIn this note we give a rather simple proof of the A 2 conjecture recently settled by T. Hytönen [7]. Theorem 1.1. For any w A 2,
A SIMPLE PROOF OF THE A 2 CONJECTURE ANDREI K. LERNER Abstract. We give a simple proof of the A 2 conecture proved recently by T. Hytönen. Our proof avoids completely the notion of the Haar shift operator,
More informationMaximal Functions in Analysis
Maximal Functions in Analysis Robert Fefferman June, 5 The University of Chicago REU Scribe: Philip Ascher Abstract This will be a self-contained introduction to the theory of maximal functions, which
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More informationRegularizations of Singular Integral Operators (joint work with C. Liaw)
1 Outline Regularizations of Singular Integral Operators (joint work with C. Liaw) Sergei Treil Department of Mathematics Brown University April 4, 2014 2 Outline 1 Examples of Calderón Zygmund operators
More informationNotes. 1 Fourier transform and L p spaces. March 9, For a function in f L 1 (R n ) define the Fourier transform. ˆf(ξ) = f(x)e 2πi x,ξ dx.
Notes March 9, 27 1 Fourier transform and L p spaces For a function in f L 1 (R n ) define the Fourier transform ˆf(ξ) = f(x)e 2πi x,ξ dx. Properties R n 1. f g = ˆfĝ 2. δλ (f)(ξ) = ˆf(λξ), where δ λ f(x)
More informationRemarks on the Gauss-Green Theorem. Michael Taylor
Remarks on the Gauss-Green Theorem Michael Taylor Abstract. These notes cover material related to the Gauss-Green theorem that was developed for work with S. Hofmann and M. Mitrea, which appeared in [HMT].
More informationON HÖRMANDER S CONDITION FOR SINGULAR INTEGRALS
EVISTA DE LA UNIÓN MATEMÁTICA AGENTINA Volumen 45, Número 1, 2004, Páginas 7 14 ON HÖMANDE S CONDITION FO SINGULA INTEGALS M. LOENTE, M.S. IVEOS AND A. DE LA TOE 1. Introduction In this note we present
More informationDecreasing Rearrangement and Lorentz L(p, q) Spaces
Decreasing Rearrangement and Lorentz L(p, q) Spaces Erik Kristiansson December 22 Master Thesis Supervisor: Lech Maligranda Department of Mathematics Abstract We consider the decreasing rearrangement of
More informationNECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES
NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)
More informationFractional integral operators on generalized Morrey spaces of non-homogeneous type 1. I. Sihwaningrum and H. Gunawan
Fractional integral operators on generalized Morrey spaces of non-homogeneous type I. Sihwaningrum and H. Gunawan Abstract We prove here the boundedness of the fractional integral operator I α on generalized
More informationMaximal functions and the control of weighted inequalities for the fractional integral operator
Indiana University Mathematics Journal, 54 (2005), 627 644. Maximal functions and the control of weighted inequalities for the fractional integral operator María J. Carro Carlos Pérez Fernando Soria and
More informationHardy-Littlewood maximal operator in weighted Lorentz spaces
Hardy-Littlewood maximal operator in weighted Lorentz spaces Elona Agora IAM-CONICET Based on joint works with: J. Antezana, M. J. Carro and J. Soria Function Spaces, Differential Operators and Nonlinear
More informationDuality of multiparameter Hardy spaces H p on spaces of homogeneous type
Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationGeometric intuition: from Hölder spaces to the Calderón-Zygmund estimate
Geometric intuition: from Hölder spaces to the Calderón-Zygmund estimate A survey of Lihe Wang s paper Michael Snarski December 5, 22 Contents Hölder spaces. Control on functions......................................2
More informationChapter 6. Integration. 1. Integrals of Nonnegative Functions. a j µ(e j ) (ca j )µ(e j ) = c X. and ψ =
Chapter 6. Integration 1. Integrals of Nonnegative Functions Let (, S, µ) be a measure space. We denote by L + the set of all measurable functions from to [0, ]. Let φ be a simple function in L +. Suppose
More informationExercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1.
Real Variables, Fall 2014 Problem set 3 Solution suggestions xercise 1. Let f be a nonnegative measurable function. Show that f = sup ϕ, where ϕ is taken over all simple functions with ϕ f. For each n
More informationRecall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm
Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationWeighted norm inequalities for singular integral operators
Weighted norm inequalities for singular integral operators C. Pérez Journal of the London mathematical society 49 (994), 296 308. Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid,
More informationMEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich
MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 9 September 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014.
More informationOn the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms
On the Brezis and Mironescu conjecture concerning a Gagliardo-Nirenberg inequality for fractional Sobolev norms Vladimir Maz ya Tatyana Shaposhnikova Abstract We prove the Gagliardo-Nirenberg type inequality
More informationAnalytic families of multilinear operators
Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia 17-20 October 2017 Based on a joint work with Loukas Grafakos M.
More informationCHAPTER 6. Differentiation
CHPTER 6 Differentiation The generalization from elementary calculus of differentiation in measure theory is less obvious than that of integration, and the methods of treating it are somewhat involved.
More informationLebesgue s Differentiation Theorem via Maximal Functions
Lebesgue s Differentiation Theorem via Maximal Functions Parth Soneji LMU München Hütteseminar, December 2013 Parth Soneji Lebesgue s Differentiation Theorem via Maximal Functions 1/12 Philosophy behind
More informationON ORLICZ-SOBOLEV CAPACITIES
ON ORLICZ-SOBOLEV CAPACITIES JANI JOENSUU Academic dissertation To be presented for public examination with the permission of the Faculty of Science of the University of Helsinki in S12 of the University
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ.
More informationGood Lambda Inequalities and Riesz Potentials for Non Doubling Measures in R n
Good Lambda Inequalities and Riesz Potentials for Non Doubling Measures in R n Mukta Bhandari mukta@math.ksu.edu Advisor Professor Charles Moore Department of Mathematics Kansas State University Prairie
More informationTHE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)
Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev
More informationMATH MEASURE THEORY AND FOURIER ANALYSIS. Contents
MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure
More informationErratum to Multipliers and Morrey spaces.
Erratum to Multipliers Morrey spaces. Pierre Gilles Lemarié Rieusset Abstract We correct the complex interpolation results for Morrey spaces which is false for the first interpolation functor of Calderón,
More informationSELF-IMPROVEMENT OF UNIFORM FATNESS REVISITED
SELF-IMPROVEMENT OF UNIFORM FATNESS REVISITED JUHA LEHRBÄCK, HELI TUOMINEN, AND ANTTI V. VÄHÄKANGAS Abstract. We give a new proof for the self-improvement of uniform p-fatness in the setting of general
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More informationBoth these computations follow immediately (and trivially) from the definitions. Finally, observe that if f L (R n ) then we have that.
Lecture : One Parameter Maximal Functions and Covering Lemmas In this first lecture we start studying one of the basic and fundamental operators in harmonic analysis, the Hardy-Littlewood maximal function.
More informationIt follows from the above inequalities that for c C 1
3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1
More informationHeat Flows, Geometric and Functional Inequalities
Heat Flows, Geometric and Functional Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France heat flow and semigroup interpolations Duhamel formula (19th century) pde, probability, dynamics
More informationNontangential limits and Fatou-type theorems on post-critically finite self-similar sets
Nontangential limits and on post-critically finite self-similar sets 4th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Universidad de Colima Setting Boundary limits
More informationHardy-Stein identity and Square functions
Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC 217 1 /
More informationV. CHOUSIONIS AND X. TOLSA
THE T THEOEM V. CHOUSIONIS AND X. TOLSA Introduction These are the notes of a short course given by X. Tolsa at the Universitat Autònoma de Barcelona between November and December of 202. The notes have
More informationBrunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian M. Novaga, B. Ruffini January 13, 2014 Abstract We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski
More informationCHAPTER I THE RIESZ REPRESENTATION THEOREM
CHAPTER I THE RIESZ REPRESENTATION THEOREM We begin our study by identifying certain special kinds of linear functionals on certain special vector spaces of functions. We describe these linear functionals
More informationFunction spaces with variable exponents
Function spaces with variable exponents Henning Kempka September 22nd 2014 September 22nd 2014 Henning Kempka 1 / 50 http://www.tu-chemnitz.de/ Outline 1. Introduction & Motivation First motivation Second
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationFUNCTION SPACES WITH VARIABLE EXPONENTS AN INTRODUCTION. Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano. Received September 18, 2013
Scientiae Mathematicae Japonicae Online, e-204, 53 28 53 FUNCTION SPACES WITH VARIABLE EXPONENTS AN INTRODUCTION Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano Received September 8, 203 Abstract. This
More informationOn the p-laplacian and p-fluids
LMU Munich, Germany Lars Diening On the p-laplacian and p-fluids Lars Diening On the p-laplacian and p-fluids 1/50 p-laplacian Part I p-laplace and basic properties Lars Diening On the p-laplacian and
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationOptimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces
Optimal embeddings of Bessel-potential-type spaces into generalized Hölder spaces J. S. Neves CMUC/University of Coimbra Coimbra, 15th December 2010 (Joint work with A. Gogatishvili and B. Opic, Mathematical
More informationHIGHER INTEGRABILITY WITH WEIGHTS
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 355 366 HIGHER INTEGRABILITY WITH WEIGHTS Juha Kinnunen University of Jyväskylä, Department of Mathematics P.O. Box 35, SF-4351
More informationSHARP L p WEIGHTED SOBOLEV INEQUALITIES
Annales de l Institut de Fourier (3) 45 (995), 6. SHARP L p WEIGHTED SOBOLEV INEUALITIES Carlos Pérez Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain e mail: cperezmo@ccuam3.sdi.uam.es
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationReview of measure theory
209: Honors nalysis in R n Review of measure theory 1 Outer measure, measure, measurable sets Definition 1 Let X be a set. nonempty family R of subsets of X is a ring if, B R B R and, B R B R hold. bove,
More information(b) If f L p (R), with 1 < p, then Mf L p (R) and. Mf L p (R) C(p) f L p (R) with C(p) depending only on p.
Lecture 3: Carleson Measures via Harmonic Analysis Much of the argument from this section is taken from the book by Garnett, []. The interested reader can also see variants of this argument in the book
More informationThe Fourier transform and Hausdorff dimension. Pertti Mattila. Pertti Mattila. University of Helsinki. Sant Feliu de Guíxols June 15 18, 2015
University of Helsinki Sant Feliu de Guíxols June 15 18, 2015 The s-al measure H s, s 0, is defined by H s (A) = lim δ 0 H s δ (A), where, for 0 < δ, H s δ (A) = inf{ j d(e j ) s : A j E j, d(e j ) < δ}.
More informationANALYSIS OF FUNCTIONS (D COURSE - PART II MATHEMATICAL TRIPOS)
ANALYSIS OF FUNCTIONS (D COURS - PART II MATHMATICAL TRIPOS) CLÉMNT MOUHOT - C.MOUHOT@DPMMS.CAM.AC.UK Contents. Integration of functions 2. Vector spaces of functions 9 3. Fourier decomposition of functions
More informationPotential Theory on Wiener space revisited
Potential Theory on Wiener space revisited Michael Röckner (University of Bielefeld) Joint work with Aurel Cornea 1 and Lucian Beznea (Rumanian Academy, Bukarest) CRC 701 and BiBoS-Preprint 1 Aurel tragically
More informationIsoperimetric Inequalities and Applications Inegalităţi Izoperimetrice şi Aplicaţii
Isoperimetric Inequalities and Applications Inegalităţi Izoperimetrice şi Aplicaţii Bogoşel Beniamin Coordonator: Prof. dr. Bucur Dorin Abstract This paper presents in the beginning the existence of the
More informationAliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide
aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle
More informationHarmonic Analysis Homework 5
Harmonic Analysis Homework 5 Bruno Poggi Department of Mathematics, University of Minnesota November 4, 6 Notation Throughout, B, r is the ball of radius r with center in the understood metric space usually
More informationRiesz Representation Theorems
Chapter 6 Riesz Representation Theorems 6.1 Dual Spaces Definition 6.1.1. Let V and W be vector spaces over R. We let L(V, W ) = {T : V W T is linear}. The space L(V, R) is denoted by V and elements of
More informationHardy spaces of Dirichlet series and function theory on polydiscs
Hardy spaces of Dirichlet series and function theory on polydiscs Kristian Seip Norwegian University of Science and Technology (NTNU) Steinkjer, September 11 12, 2009 Summary Lecture One Theorem (H. Bohr)
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationMathematical Research Letters 4, (1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS. Juha Kinnunen and Olli Martio
Mathematical Research Letters 4, 489 500 1997) HARDY S INEQUALITIES FOR SOBOLEV FUNCTIONS Juha Kinnunen and Olli Martio Abstract. The fractional maximal function of the gradient gives a pointwise interpretation
More informationThe Hardy Operator and Boyd Indices
The Hardy Operator and Boyd Indices Department of Mathematics, University of Mis- STEPHEN J MONTGOMERY-SMITH souri, Columbia, Missouri 65211 ABSTRACT We give necessary and sufficient conditions for the
More informationCONDUCTOR SOBOLEV TYPE ESTIMATES AND ISOCAPACITARY INEQUALITIES
CONDUCTOR SOBOLEV TYPE ESTIMATES AND ISOCAPACITARY INEQUALITIES JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Abstract. In this paper we present an integral inequality connecting a function space (quasi-)norm
More informationCAPACITIES ON METRIC SPACES
[June 8, 2001] CAPACITIES ON METRIC SPACES V. GOL DSHTEIN AND M. TROYANOV Abstract. We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces.
More information+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1
Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and
More informationJordan Journal of Mathematics and Statistics (JJMS) 9(1), 2016, pp BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT
Jordan Journal of Mathematics and Statistics (JJMS 9(1, 2016, pp 17-30 BOUNDEDNESS OF COMMUTATORS ON HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT WANG HONGBIN Abstract. In this paper, we obtain the boundedness
More informationCHARACTERIZATION OF ORLICZ-SOBOLEV SPACE
CHRCTERIZTION OF ORLICZ-SOBOLEV SPCE HELI TUOMINEN bstract. We give a new characterization of the Orlicz-Sobolev space W 1,Ψ (R n ) in terms of a pointwise inequality connected to the Young function Ψ.
More informationAnalysis Comprehensive Exam Questions Fall 2008
Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationA SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM
A SHORT PROOF OF THE COIFMAN-MEYER MULTILINEAR THEOREM CAMIL MUSCALU, JILL PIPHER, TERENCE TAO, AND CHRISTOPH THIELE Abstract. We give a short proof of the well known Coifman-Meyer theorem on multilinear
More informationCONTENTS. 4 Hausdorff Measure Introduction The Cantor Set Rectifiable Curves Cantor Set-Like Objects...
Contents 1 Functional Analysis 1 1.1 Hilbert Spaces................................... 1 1.1.1 Spectral Theorem............................. 4 1.2 Normed Vector Spaces.............................. 7 1.2.1
More informationA characterization of the two weight norm inequality for the Hilbert transform
A characterization of the two weight norm inequality for the Hilbert transform Ignacio Uriarte-Tuero (joint works with Michael T. Lacey, Eric T. Sawyer, and Chun-Yen Shen) September 10, 2011 A p condition
More informationANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n
ANALYSIS QUALIFYING EXAM FALL 206: SOLUTIONS Problem. Let m be Lebesgue measure on R. For a subset E R and r (0, ), define E r = { x R: dist(x, E) < r}. Let E R be compact. Prove that m(e) = lim m(e /n).
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer 1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First
More informationRESTRICTED WEAK TYPE VERSUS WEAK TYPE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 4, Pages 1075 1081 S 0002-9939(04)07791-3 Article electronically published on November 1, 2004 RESTRICTED WEAK TYPE VERSUS WEAK TYPE
More informationWeighted restricted weak type inequalities
Weighted restricted weak type inequalities Eduard Roure Perdices Universitat de Barcelona Joint work with M. J. Carro May 18, 2018 1 / 32 Introduction Abstract We review classical results concerning the
More informationON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS
ON THE ENDPOINT REGULARITY OF DISCRETE MAXIMAL OPERATORS EMANUEL CARNEIRO AND KEVIN HUGHES Abstract. Given a discrete function f : Z d R we consider the maximal operator X Mf n = sup f n m, r 0 Nr m Ω
More informationJUHA KINNUNEN. Real Analysis
JUH KINNUNEN Real nalysis Department of Mathematics and Systems nalysis, alto University Updated 3 pril 206 Contents L p spaces. L p functions..................................2 L p norm....................................
More informationFunctional Analysis, Stein-Shakarchi Chapter 1
Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises
More informationCalderón-Zygmund inequality on noncompact Riem. manifolds
The Calderón-Zygmund inequality on noncompact Riemannian manifolds Institut für Mathematik Humboldt-Universität zu Berlin Geometric Structures and Spectral Invariants Berlin, May 16, 2014 This talk is
More informationOn a class of pseudodifferential operators with mixed homogeneities
On a class of pseudodifferential operators with mixed homogeneities Po-Lam Yung University of Oxford July 25, 2014 Introduction Joint work with E. Stein (and an outgrowth of work of Nagel-Ricci-Stein-Wainger,
More informationIntegration on Measure Spaces
Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationAPPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( )
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 200, 405 420 APPROXIMATE IDENTITIES AND YOUNG TYPE INEQUALITIES IN VARIABLE LEBESGUE ORLICZ SPACES L p( ) (log L) q( ) Fumi-Yuki Maeda, Yoshihiro
More informationNEW MAXIMAL FUNCTIONS AND MULTIPLE WEIGHTS FOR THE MULTILINEAR CALDERÓN-ZYGMUND THEORY
NEW MAXIMAL FUNCTIONS AND MULTIPLE WEIGHTS FOR THE MULTILINEAR CALDERÓN-ZYGMUND THEORY ANDREI K. LERNER, SHELDY OMBROSI, CARLOS PÉREZ, RODOLFO H. TORRES, AND RODRIGO TRUJILLO-GONZÁLEZ Abstract. A multi(sub)linear
More informationOn a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
Caspian Journal of Applied Mathematics, Economics and Ecology V. 1, No 1, 2013, July ISSN 1560-4055 On a compactness criteria for multidimensional Hardy type operator in p-convex Banach function spaces
More informationFRACTIONAL HARDY INEQUALITIES AND VISIBILITY OF THE BOUNDARY
FRACTIONAL HARDY INEQUALITIES AND VISIBILITY OF THE BOUNDARY LIZAVETA IHNATSYEVA, JUHA LEHRBÄCK, HELI TUOMINEN, AND ANTTI V. VÄHÄKANAS Abstract. We prove fractional order Hardy inequalities on open sets
More information