Capacitary Riesz-Herz and Wiener-Stein estimates

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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.

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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.