u(x,t)= í /(e^wv^dí.
|
|
- Loreen Edwards
- 1 years ago
- Views:
Transcription
1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 4, April 1988 SCHRODINGER EQUATIONS: POINTWISE CONVERGENCE TO THE INITIAL DATA (Communicated LUIS VEGA by Richard R. Goldberg) ABSTRACT. Let u(x,i) be the solution of the Schrödinger equation with initial data / in the Sobolev space ifs(rn) with a > t. The a.e. convergence of u(x, i) to f(x) follows from a weighted estimate of the maximal function u*(x,t) = supi>0 u(x,t). 0. Let / be in the Schwartz space S(R") and for x e R", í R set u(x,t)= í /(e^wv^dí. ÍR" It is well known that u is the solution of the Schrödinger data /. Au = i-u, t > 0, u(x,0) = f(x). equation with initial For s R we denote by Hs(Rn) the Sobolev space H9(Rn) = if S'(R") s.t. H/IU^r.) = ( (1 + e 3)*l/(0l2) < We obtain the following result. THEOREM 1. Let f be in Hs{Rn) with s> \. Then lim u(x,t) = f(x) a.e. x. This result is a consequence of the boundedness of the maximal operator u* (x) = sup0<lti\u(x,t)\. THEOREM 2. Let f be in Hs{Rn) with s > a/2 and a > 1. Then (/' Since HT(R) with r > is embedded in L (R) by the classical Sobolev inequalities, Theorem 2 is an immediate consequence of THEOREM 3. IfO<a f r roc,jr" J -ao - u(x,t) dta y ' anda > 1, then (i'x dt(\ + \x\)a 1 ^Cll/H//^-' +»/2(R")- Received by the editors December 4, Mathematics Subject Classification (1985 Revision). Primary 42A45; Secondary 42B American Mathematical Society /88 $ $.25 per page
2 SCHRÖDINGER EQUATIONS 875 The above results are best known for n > 3. For n = 1, L. Carleson [2] proved the a.e. convergence for / e Hx^{Rn) and constructed an example of an / Hl^{R) such that u(x,t) does not converge to / a.e. Later, B. Dahlberg and C. Kenig [5] proved that the positive result of Carleson was sharp. These questions and the boundedness of the maximal operator have also been studied by C. Kenig and A. Ruiz [6], A. Carbery [1] and M. Cowling [4]. They prove that if / #n/4(rn) or / G Hs{Rn) for s > 1, then the maximal function is bounded. We also give an alternative proof to the fact that there is no boundedness for the maximal operator, and then no convergence result, for / Hs(Rn) with s < (!) In 2, we generalize the above theorems to multipliers of the type e1'^ '. These results have been obtained at the same time and independently by P. Sjölin. We should like to thank S. Córdoba for his help and encouragement. 1. Proof of Theorem 3. We shall make use of the following inequalitiy which is proved below. LEMMA. Let g be in L2(Sn~1). Then if a > 1 / / g(ç)eix-t da(t) Jr." Js"-' ;i + x )«<c CL'H u{x. t) = \ H elstsn-2^ i f{sl'20e%su2xi da(0 ds, 2 jo y.s"-' and then we use Plancherel's inequality in the t variable to get / f Jr." J-, in polar coordi- With a simple change of variable we obtain the representation nates, da - u(x,t) dt- dta ' l + \x\ L a + (n-2)/2 4 Ir" Jo f(s^2^e"i/2^da(0 ds- l+i " <- f (1 + s)<*/2 + n/2 + 2a-2 Í f(s l/2\ ix- Mt) ds, 4 JO JR" 1 + \x\ by the lemma r f 2 <C (1 + s)a/2 + n/2 + 2a-2 / ffav*ç) da(0 Jo Js"~l c\\f\\h2n-' + a'2(rn)- For n = 1 the proof is similar. In this case one proves I - uix, t) dta,2 X dt\ IZ,o (R) ds < c //-' '(R) D PROOF OF THE LEMMA. We want to see that L2(Sn~1) is embedded in H~s(Rn) with s > k. Though this fact is an immediate consequence of the trace
3 876 LUIS VEGA theorem, we give an alternative proof. We shall do this in S1. The generalization to higher dimensions is straightforward. It is sufficient to prove the inequalitiy for simple functions having their supports contained in {x = el6, 0 < 6 < n/4}. Set 6 > 0 and consider for every R} = J x - elf>es'-.j- <0<{j + l)- ;-i/2] j e {0,..., [<5-]}. Each Rj can be covered by a rectangle Rj of dimensions c\6 x C26 in the normal and tangent directions to Sl at the point el^w^4^s D where ci, ci are convenient universal constants greater than 1 (see the figure). Then, it is sufficient to show 2 V/2 / \ L??** where {<Pj} is a smooth partition of unity subordinate to {Rj} and a3 C. But /R2iEwr(i+ixi)-</E^(1+liil)a/2 2 -/ ^ajlpj *M where / (x) = 1/(1 + :ri )a/2. Then t(0 is a measure which acts only on the 1 direction and by the finite overlapping of the supports of <pj, / 2 Sai(Pj 2 M <2Elajl2/ l^ * H Therefore, it is sufficient to see that c5/2
4 SCHRÖDINGER EQUATIONS 877 If we call Tj and r\3 the tangent and normal directions in e'w4!* ] we can take <Pj such that Then, integrating j by parts, <cè~a'2, drâ <p,(0<c6- \<Pj{*)\ = \j vm^** < c2~2k83'2 for k = 0,1,... and x = xxtj + x2r j\ 2kè~l < xx < 2k+16~1; 2k6~1'2 < x2 < 2k+\fi-i/2 Since we always have v2;(z) < c3/2, then / ^ 2(l + x1 )-a/2<c^2-3fc-+3. D * 2. Negative results. THEOREM 4. The inequality Je does not hold for s <. B(0,R) \u*{x)\2\ k <C«/ «.(R») PROOF. Let <pk e Cg ([2fc,2* + 2fc/2]) with k = 0,1,... and 0 < <j>k < 1, \{dl/l)<t>k{x)\ < C2-{-kl^\ It is known that for [a;] sufficiently large Then - i i {c1e*'il+cae-"atl} + o L.-rid ^-mr-w la-kn+i) uk(x,t)= [ e^2tm0elxido{0 JR" = _Î_ [ ir' I (n-l)/2 J t-'wrln-wfatfdr + pt(r2t + r x )r(n-l)/2 la-lin-ij/a' <f>k(r)dr = ^W + ^W + ^( T)720(2fe(("/2,-1))- For every x R such that < \x\ < 1 we choose t N 1 2fc+1 + 2<fc/2' + 1 ' Let us define /?i(r) = r2^ - r i. Then r '1{2k + 2kl2 2) = 0 and by the stationary phase lemma we can assure h{x) > (c/ i (n_1^2)2fc(n/2'. Let r 2{r) = r2tx + r\x\. Then rj^r) > ^ and integrating by parts -i2(z) < c. 2*((n/2)-i). Thcn.(j.) > C2K"/2) for < s < 1. Since \\pk\\h'(r") < c2*(s+n/2-i/4) we conc]ucje that s < I is necessary. G
5 878 LUIS VEGA FURTHER RESULTS. These theorems can be generalized to multipliers of the type e'l^l '. In the following we state the results we have obtained using the techniques presented above. Let us define ub{x,t) = /(cde^'v1«do{ ). THEOREM 1'. For s> and f G Hs(Rn) lim Uh(x.t) fix) a.e. x. i->0 THEOREM 2'. Let u*b{x) =supt \ub(x,t)\ and a > 1. Then (/rj":(i)!(ittííf)"2 C i/ii'" r"',0">ar THEOREM 3'. For a > 1 and a > 0 dn ;Ub{x,t dtc V/2 dtl-, j_m\a - C / ii(a-l/2)i, + a/2(r ). (l + l^l)a THEOREM 4'. Let b > 1. The inequality \Jb{o,r) does not hold for s < \. 2 \ ( sup ií6(j,í) ) ) < Cßll/Htf^R-) \0<i<l / / REFERENCES 1. A. Carbery, Radial Fourier multipliers and associated maximal functions, Recent Progress in Fourier Analysis, North-Holland Mathematics Studies, North-Holland, Amsterdam, p L. Carleson, Some analytical problems related to statistical mechanics, Euclidean Harmonic Analysis, Lecture Notes in Math., vol. 779, Springer-Verlag, Berlin and New York, 1979, pp A. Córdoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), M. Cowling, Pointwise behaviour of solutions to Schrodinger equations, Harmonic Analysis, Lecture Notes in Math., vol. 992, Springer-Verlag, Berlin and New York, 1983, pp B. Dahlberg and C. Kenig, A note on the almost everywhere behaviour of solutions to the Schrodinger equation, Harmonic Analysis, Lecture Notes in Math., vol. 908, Springer-Verlag, Berlin and New York, 1982, pp C. Kenig and A. Ruiz, A strong type (2,2) estimate for a maximal operator associated to the Schrodinger equation, Trans. Amer. Math. Soc. 280 (1983), P. Sjölin, Regularity of solutions to the Schrodinger equation, Duke Math. J. 55 (1987), Division de Matemáticas, Universidad Autónoma, Madrid, Spain
Schrödinger equation on the Heisenberg group
STUDIA MATHEMATICA 161 (2) (2004) Schrödinger equation on the Heisenberg group by Jacek Zienkiewicz (Wrocław) Abstract. Let L be the full laplacian on the Heisenberg group H n of arbitrary dimension n.
SHARP 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
ON AN INEQUALITY OF KOLMOGOROV AND STEIN
BULL. AUSTRAL. MATH. SOC. VOL. 61 (2000) [153-159] 26B35, 26D10 ON AN INEQUALITY OF KOLMOGOROV AND STEIN HA HUY BANG AND HOANG MAI LE A.N. Kolmogorov showed that, if /,/',..., /'"' are bounded continuous
SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS
SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS PETER L. DUREN Riesz products are a useful apparatus for constructing singular functions with special properties. They have been an important source
PARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION
A PHYSICAL SPACE PROOF OF THE BILINEAR STRICHARTZ AND LOCAL SMOOTHING ESTIMATES FOR THE SCHRÖDINGER EQUATION TERENCE TAO Abstract. Let d 1, and let u, v : R R d C be Schwartz space solutions to the Schrödinger
SQUARE FUNCTION ESTIMATES AND THE T'b) THEOREM STEPHEN SEMMES. (Communicated by J. Marshall Ash)
proceedings of the american mathematical society Volume 110, Number 3, November 1990 SQUARE FUNCTION ESTIMATES AND THE T'b) THEOREM STEPHEN SEMMES (Communicated by J. Marshall Ash) Abstract. A very simple
(2) * ; v(x,y)=r-j [p(x + ty)~ p(x-ty)]dt (r > 0)
proceedings of the american mathematical society Volume 100, Number 4, August 1987 ON THE DILATATION ESTIMATES FOR BEURLING-AHLFORS QUASICONFORMAL EXTENSION DELIN TAN ABSTRACT. Let ß(x) be a p-quasisymmetric
Endpoint Strichartz estimates for the magnetic Schrödinger equation
Journal of Functional Analysis 58 (010) 37 340 www.elsevier.com/locate/jfa Endpoint Strichartz estimates for the magnetic Schrödinger equation Piero D Ancona a, Luca Fanelli b,,luisvega b, Nicola Visciglia
SOME PROPERTIES OF CONJUGATE HARMONIC FUNCTIONS IN A HALF-SPACE
SOME PROPERTIES OF CONJUGATE HARMONIC FUNCTIONS IN A HALF-SPACE ANATOLY RYABOGIN AND DMITRY RYABOGIN Abstract. We prove a multi-dimensional analog of the Theorem of Hard and Littlewood about the logarithmic
HARMONIC 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
AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0
AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional
A Remark on the Regularity of Solutions of Maxwell s Equations on Lipschitz Domains
A Remark on the Regularity of Solutions of Maxwell s Equations on Lipschitz Domains Martin Costabel Abstract Let u be a vector field on a bounded Lipschitz domain in R 3, and let u together with its divergence
ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
. A NOTE ON THE RESTRICTION THEOREM AND GEOMETRY OF HYPERSURFACES
. A NOTE ON THE RESTRICTION THEOREM AND GEOMETRY OF HYPERSURFACES FABIO NICOLA Abstract. A necessary condition is established for the optimal (L p, L 2 ) restriction theorem to hold on a hypersurface S,
A NOTE ON A BASIS PROBLEM
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 2, September 1975 A NOTE ON A BASIS PROBLEM J. M. ANDERSON ABSTRACT. It is shown that the functions {exp xvx\v_. form a basis for the
BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS
BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS BY ELIAS M. STEIN Communicated by J. J. Kohn, May 18, 1970 Let 2D be a bounded domain in C n with smooth boundary. We shall consider the behavior near the boundary
Three hours THE UNIVERSITY OF MANCHESTER. 24th January
Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the
UNIQUENESS OF THE UNIFORM NORM
proceedings of the american mathematical society Volume 116, Number 2, October 1992 UNIQUENESS OF THE UNIFORM NORM WITH AN APPLICATION TO TOPOLOGICAL ALGEBRAS S. J. BHATT AND D. J. KARIA (Communicated
THE POISSON TRANSFORM^)
THE POISSON TRANSFORM^) BY HARRY POLLARD The Poisson transform is defined by the equation (1) /(*)=- /" / MO- T J _M 1 + (X t)2 It is assumed that a is of bounded variation in each finite interval, and
SOME HILBERT SPACE EXTREMAL PROBLEMS
SOME HILBERT SPACE EXTREMAL PROBLEMS MARVIN ROSENBLUM1 In this note we shall generalize in several ways the following theorem of Philip Davis: Let I2 be a real or complex sequential Hubert space. Fix \hj\o
Some lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
A RECONSTRUCTION FORMULA FOR BAND LIMITED FUNCTIONS IN L 2 (R d )
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3593 3600 S 0002-9939(99)04938-2 Article electronically published on May 6, 1999 A RECONSTRUCTION FORMULA FOR AND LIMITED FUNCTIONS
SINGULAR INTEGRALS ON SIERPINSKI GASKETS
SINGULAR INTEGRALS ON SIERPINSKI GASKETS VASILIS CHOUSIONIS Abstract. We construct a class of singular integral operators associated with homogeneous Calderón-Zygmund standard kernels on d-dimensional,
ALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES
Electronic Journal of Differential Equations, Vol. 21(21, No. 72, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ALMOST PERIODIC SOLUTIONS
GRAND 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
WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE
WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNTIONS OF EXPONENTIAL TYPE DORON S. LUBINSKY A. We prove eighted Markov-Bernstein inequalities of the form f x p x dx σ + p f x p x dx Here satisfies
ON THE KAKEYA SET CONJECTURE
ON THE KAKEYA SET CONJECTURE J.ASPEGREN Abstract. In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture
Friedrich symmetric systems
viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary
Moore Penrose inverses and commuting elements of C -algebras
Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We
Weighted Littlewood-Paley inequalities
Nicolaus Copernicus University, Toruń, Poland 3-7 June 2013, Herrnhut, Germany for every interval I in R let S I denote the partial sum operator, i.e., (S I f ) = χ I f (f L2 (R)), for an arbitrary family
ERGODIC PATH PROPERTIES OF PROCESSES WITH STATIONARY INCREMENTS
J. Austral. Math. Soc. 72 (2002, 199 208 ERGODIC ATH ROERTIES OF ROCESSES WITH STATIONARY INCREMENTS OFFER KELLA and WOLFGANG STADJE (Received 14 July 1999; revised 7 January 2001 Communicated by V. Stefanov
DEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen
(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
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 95, Number 3, November 1985 COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR SHIGERU SAKAGUCHI Abstract. We consider the obstacle
TRANSFORMATION FORMULA OF HIGHER ORDER INTEGRALS
J. Austral. Math. Soc. (Series A) 68 (2000), 155-164 TRANSFORMATION FORMULA OF HIGHER ORDER INTEGRALS TAO QIAN and TONGDE ZHONG (Received 21 December 1998; revised 30 June 1999) Communicated by P. G. Fenton
ON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS
J. Austral. Math. Soc. (Series A) 43 (1987), 279-286 ON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS WOJC3ECH KUCHARZ (Received 15 April 1986) Communicated by J. H. Rubinstein Abstract
ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
ON THE BEHAVIOR OF THE SOLUTION OF THE WAVE EQUATION. 1. Introduction. = u. x 2 j
ON THE BEHAVIO OF THE SOLUTION OF THE WAVE EQUATION HENDA GUNAWAN AND WONO SETYA BUDHI Abstract. We shall here study some properties of the Laplace operator through its imaginary powers, and apply the
A 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
ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS
J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.
Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures
2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon
COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
ON FINITE INVARIANT MEASURES FOR MARKOV OPERATORS1 M. FALKOWITZ
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 3, May 1973 ON FINITE INVARIANT MEASURES FOR MARKOV OPERATORS1 M. FALKOWITZ Abstract. Two lemmas on proper vectors of convex linear combination
HARNACK INEQUALITY FOR NONDIVERGENT ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS. Seick Kim
HARNACK INEQUALITY FOR NONDIVERGENT ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS Seick Kim We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian
THE SPECTRAL DIAMETER IN BANACH ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume»!. Number 1, Mav 1984 THE SPECTRAL DIAMETER IN BANACH ALGEBRAS SANDY GRABINER1 Abstract. The element a is in the center of the Banach algebra A modulo
A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (2010),
A G Ramm, Implicit Function Theorem via the DSM, Nonlinear Analysis: Theory, Methods and Appl., 72, N3-4, (21), 1916-1921. 1 Implicit Function Theorem via the DSM A G Ramm Department of Mathematics Kansas
SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
MAJORIZING MEASURES WITHOUT MEASURES. By Michel Talagrand URA 754 AU CNRS
The Annals of Probability 2001, Vol. 29, No. 1, 411 417 MAJORIZING MEASURES WITHOUT MEASURES By Michel Talagrand URA 754 AU CNRS We give a reformulation of majorizing measures that does not involve measures,
Null-controllability of the heat equation in unbounded domains
Chapter 1 Null-controllability of the heat equation in unbounded domains Sorin Micu Facultatea de Matematică-Informatică, Universitatea din Craiova Al. I. Cuza 13, Craiova, 1100 Romania sd micu@yahoo.com
Generalized Ward and Related Clustering Problems
Generalized Ward and Related Clustering Problems Vladimir BATAGELJ Department of mathematics, Edvard Kardelj University, Jadranska 9, 6 000 Ljubljana, Yugoslavia Abstract In the paper an attempt to legalize
ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER. Communicated by W. Feller, May 1, 1962
ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER Communicated by W. Feller, May 1, 1962 1. Let x(t) with oo
RINGS WITH UNIQUE ADDITION
RINGS WITH UNIQUE ADDITION Dedicated R. E. JOHNSON to the Memory of Tibor Szele Introduction. The ring {R; +, } is said to have unique addition if there exists no other ring {R; +', } having the same multiplicative
ON THE SIZE OF KAKEYA SETS IN FINITE FIELDS
ON THE SIZE OF KAKEYA SETS IN FINITE FIELDS ZEEV DVIR Abstract. A Kakeya set is a subset of F n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show
1 Stochastic Dynamic Programming
1 Stochastic Dynamic Programming Formally, a stochastic dynamic program has the same components as a deterministic one; the only modification is to the state transition equation. When events in the future
Notes. 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)
ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
Optimal 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
The initial value problem for non-linear Schrödinger equations
The initial value problem for non-linear Schrödinger equations LUIS VEGA Universidad del País Vasco Euskal Herriko Unibertsitatea ICM MADRID 2006 1 In this talk I shall present some work done in collaboration
Hardy-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
MATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
238 H. S. WILF [April
238 H. S. WILF [April 2. V. Garten, Über Tauber'she Konstanten bei Cesaro'sehen Mittlebildung, Comment. Math. Helv. 25 (1951), 311-335. 3. P. Hartman, Tauber's theorem and absolute constants, Amer. J.
RESEARCH ANNOUNCEMENTS
RESEARCH ANNOUNCEMENTS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 24, Number 2, April 1991 DISTRIBUTION RIGIDITY FOR UNIPOTENT ACTIONS ON HOMOGENEOUS SPACES MARINA RATNER In this
NOTE ON THE DERIVATIVES OF FUNCTIONS ANALYTIC IN THE UNIT CIRCLE J. L. WALSH
NOTE ON THE DERIVATIVES OF FUNCTIONS ANALYTIC IN THE UNIT CIRCLE J. L. WALSH In a recent paper, 1 W. Seidel and the present writer established distortion theorems for various classes of functions analytic
(2) ^max^lpm g M/?"[l - ^-(l - /T1)2}
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 56, April 1976 SOME INEQUALITIES FOR POLYNOMIALS Q. I. RAHMAN Abstract. Let pn(z) be a polynomial of degree n. Given that pn(z) has a zero on the
Binomial coefficient codes over W2)
Discrete Mathematics 106/107 (1992) 181-188 North-Holland 181 Binomial coefficient codes over W2) Persi Diaconis Harvard University, Cambridge, MA 02138, USA Ron Graham AT & T Bell Laboratories, Murray
New York Journal of Mathematics. A Refinement of Ball s Theorem on Young Measures
New York Journal of Mathematics New York J. Math. 3 (1997) 48 53. A Refinement of Ball s Theorem on Young Measures Norbert Hungerbühler Abstract. For a sequence u j : R n R m generating the Young measure
NON-COMPACT COMPOSITION OPERATORS
BULL. AUSTRAL. MATH. SOC. 47B99 VOL. 21 (1980), 125-130. (46JI5) NON-COMPACT COMPOSITION OPERATORS R.K. SINGH AND S.D. SHARMA In this note sufficient conditions for non-compactness of composition operators
SLICING THE CUBE IN R" AND PROBABILITY (BOUNDS FOR THE MEASURE OF A CENTRAL CUBE SLICE IN R" BY PROBABILITY METHODS)
AMERICAN MATHEMATICAL SOCIETY Volume 73, Number 1, January 1979 SLICING THE CUBE IN R" AND PROBABILITY (BOUNDS FOR THE MEASURE OF A CENTRAL CUBE SLICE IN R" BY PROBABILITY METHODS) DOUGLAS HENSLEY Abstract.
du(y)= I e~ix'y du(x), y R".
proceedings of the american mathematical society Volume 118, Number 4, August 1993 THE NULL SET OF THE FOURIER TRANSFORM FOR A SURFACE CARRIED MEASURE LIMIN SUN (Communicated by J. Marshall Ash) Abstract.
THE ISOENERGY INEQUALITY FOR A HARMONIC MAP
HOUSTON JOURNAL OF MATHEMATICS @ 1998 University of Houston Volume 24, No. 4, 1998 THE ISOENERGY INEQUALITY FOR A HARMONIC MAP JAIGYOUNG CHOE Communicated by Robert M. Hardt ABSTRACT. Let u be a harmonic
A local estimate from Radon transform and stability of Inverse EIT with partial data
A local estimate from Radon transform and stability of Inverse EIT with partial data Alberto Ruiz Universidad Autónoma de Madrid ge Joint work with P. Caro (U. Helsinki) and D. Dos Santos Ferreira (Paris
Invariant measures and the compactness of the domain
ANNALES POLONICI MATHEMATICI LXIX.(998) Invariant measures and the compactness of the domain by Marian Jab loński (Kraków) and Pawe l Góra (Montreal) Abstract. We consider piecewise monotonic and expanding
General Mathematics Vol. 16, No. 1 (2008), A. P. Madrid, C. C. Peña
General Mathematics Vol. 16, No. 1 (2008), 41-50 On X - Hadamard and B- derivations 1 A. P. Madrid, C. C. Peña Abstract Let F be an infinite dimensional complex Banach space endowed with a bounded shrinking
AN ARCSINE LAW FOR MARKOV CHAINS
AN ARCSINE LAW FOR MARKOV CHAINS DAVID A. FREEDMAN1 1. Introduction. Suppose {xn} is a sequence of independent, identically distributed random variables with mean 0. Under certain mild assumptions [4],
A Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths
International Journal of Mathematical Analysis Vol. 9, 05, no. 48, 387-406 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.589 A Present Position-Dependent Conditional Fourier-Feynman Transform
TOOLS FROM HARMONIC ANALYSIS
TOOLS FROM HARMONIC ANALYSIS BRADLY STADIE Abstract. The Fourier transform can be thought of as a map that decomposes a function into oscillatory functions. In this paper, we will apply this decomposition
A NOTE ON POSITIVE DEFINITE NORM DEPENDENT FUNCTIONS
A NOTE ON POSITIVE DEFINITE NORM DEPENDENT FUNCTIONS ALEXANDER KOLDOBSKY Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f : [, ) R, that if
Maximal 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
THE REGULAR ELEMENT PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2123 2129 S 0002-9939(98)04257-9 THE REGULAR ELEMENT PROPERTY FRED RICHMAN (Communicated by Wolmer V. Vasconcelos)
Implicit Functions, Curves and Surfaces
Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then
A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3185 3189 S 0002-9939(97)04112-9 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by
Nonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
Fractional 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
Sectorial Forms and m-sectorial Operators
Technische Universität Berlin SEMINARARBEIT ZUM FACH FUNKTIONALANALYSIS Sectorial Forms and m-sectorial Operators Misagheh Khanalizadeh, Berlin, den 21.10.2013 Contents 1 Bounded Coercive Sesquilinear
COMMENTARY ON TWO PAPERS OF A. P. CALDERÓN
COMMENTARY ON TWO PAPERS OF A. P. CALDERÓN MICHAEL CHRIST Is she worth keeping? Why, she is a pearl Whose price hath launched above a thousand ships. (William Shakespeare, Troilus and Cressida) Let u be
FUNDAMENTAL THEOREMS OF ANALYSIS IN FORMALLY REAL FIELDS
Novi Sad J. Math. Vol. 8, No., 2008, 27-2 FUNDAMENTAL THEOREMS OF ANALYSIS IN FORMALLY REAL FIELDS Angelina Ilić Stepić 1, Žarko Mijajlović 2 Abstract. The theory of real closed fields admits elimination
ON KANNAN MAPS. CHI SONG WONGl. ABSTRACT. Let K be a (nonempty) weakly compact convex subset of
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 1, January 1975 ON KANNAN MAPS CHI SONG WONGl ABSTRACT. Let K be a (nonempty) weakly compact convex subset of a Banach space B. Let T
On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous
HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 HERMITE MULTIPLIERS AND PSEUDO-MULTIPLIERS JAY EPPERSON Communicated by J. Marshall Ash) Abstract. We prove a multiplier
POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES
POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES ALEXANDER KOLDOBSKY Abstract. We prove that if a function of the form f( K is positive definite on, where K is an origin symmetric
A LeVeque-type lower bound for discrepancy
reprinted from Monte Carlo and Quasi-Monte Carlo Methods 998, H. Niederreiter and J. Spanier, eds., Springer-Verlag, 000, pp. 448-458. A LeVeque-type lower bound for discrepancy Francis Edward Su Department
SUBRINGS OF NOETHERIAN RINGS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, November 1974 SUBRINGS OF NOETHERIAN RINGS EDWARD FORMANEK AND ARUN VINAYAK JATEGAONKAR ABSTRACT. Let S be a subring of a ring R such
On the exponential map on Riemannian polyhedra by Monica Alice Aprodu. Abstract
Bull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 3, 2017, 233 238 On the exponential map on Riemannian polyhedra by Monica Alice Aprodu Abstract We prove that Riemannian polyhedra admit explicit
1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
The 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}.
A PROOF OF THE TRACE THEOREM OF SOBOLEV SPACES ON LIPSCHITZ DOMAINS
POCEEDINGS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 4, Number, February 996 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES ON LIPSCHITZ DOMAINS ZHONGHAI DING (Communicated by Palle E. T. Jorgensen) Abstract.
AN Lx REMAINDER THEOREM FOR AN INTEGRODIFFERENTIAL EQUATION WITH ASYMPTOTICALLY PERIODIC SOLUTION KENNETH B. HANNSGEN
proceedings of the american mathematical society Volume 73, Number 3, March 1979 AN Lx REMAINDER THEOREM FOR AN INTEGRODIFFERENTIAL EQUATION WITH ASYMPTOTICALLY PERIODIC SOLUTION KENNETH B. HANNSGEN Abstract.
Lecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and