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Weighted Inequalities for Singular Integral Operators on the Half-Line Ralph Chill, Sebastian Król

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Weighted Inequalities for Singular Integral Operators on the Half-Line Ralph Chill, Sebastian Król Weighted inequalities for singular integral operators on the half-line Ralph Chill, Sebastian Król To cite this version: Ralph Chill, Sebastian Król. Weighted inequalities for singular integral operators on the half-line. 2016. hal-01278755 HAL Id: hal-01278755 https://hal.archives-ouvertes.fr/hal-01278755 Preprint submitted on 24 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. WEIGHTED INEQUALITIES FOR SINGULAR INTEGRAL OPERATORS ON THE HALF-LINE RALPH CHILL AND SEBASTIAN KRÓL ABSTRACT. We prove weighted estimates for singular integral operators which oper- ate on function spaces on a half-line. The class of admissible weights includes Muck- enhoupt weights and weights satisfying Sawyer’s one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal Lp regularity of first order, second order and fractional or- der Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson, and Chill and Fiorenza. 1. MOTIVATION AND DESCRIPTION OF THE MAIN RESULT Weighted Lp-estimates for singular integral operators play an important role in har- monic analysis and in its applications to elliptic and parabolic partial differential equa- tions, as well as abstract evolution equations on Banach spaces, where singular inte- gral operators with operator valued kernels naturally arise via representation formu- las for solutions. These applications are in turn important for the theory of nonlinear equations where one is often interested in obtaining wellposedness and regularity re- sults for data in various function spaces. A prominent part of the literature is devoted to weighted inequalities for the Hardy- Littlewood maximal function and singular integral operators on Rn. Motivated in p p particular by applications to L -maximal regularity, or actually Lw-maximal regular- ity and Ew-maximal regularity (E being a rearrangement invariant Banach function space), of first order, second order and fractional order Cauchy problems on the half- line, we study in this article weighted inequalities for singular integral operators on the half-line. This special situation brings in some new features which we combine with techniques which have been developed recently in the context of singular integral operators on function spaces on the line or on Rn. By concentrating on the half-line case, the natural context in the above mentioned applications, we obtain weighted inequalities for a larger class of weights, which includes the class of Muckenhoupt weights and the class of weights satisfying the one-sided Sawyer condition on the line. Second, we obtain weighted inequalities for singular integral operators with kernels which satisfy comparatively weak regularity conditions. Third, by relying on Rubio de Francia’s extrapolation technique, we obtain not only weighted Lp-estimates, but an extrapolation result in the context of rearrangement invariant Banach function spaces. Date: March 19, 2015. 2010 Mathematics Subject Classification. 46D05. Key words and phrases. singular integral operators, weighted estimates, extrapolation, rearrangement invariant Banach function space, first order Cauchy problem, maximal regularity. The second author is grateful for support by the Alexander von Humboldt Foundation. 1 2 RALPH CHILL AND SEBASTIAN KRÓL Let us explain these results in more detail and prepare the notational background, by recalling first the situation of singular integral operators on the line. Throughout, let X and Y be two Banach spaces with norms j · jX and j · jY, respectively. A measur- 1 able function K : R × R !L(X, Y) is called a kernel if K(t, ·) 2 Lloc(R n ftg; L(X, Y)) for every t 2 R. We say that a bounded linear operator T from Lp(R; X) into Lp(R; Y) (p 2 (1, ¥)) is a singular integral operator if there exists a kernel K such that Z T f (t) = K(t, s) f (s)ds R ¥ for every f 2 Lc (R; X) and every t 2/ supp f . ¥ Here, Lc (R; X) stands for the space of all X-valued, essentially bounded, measur- able functions with compact support in R. A singular integral operator with ker- nel K is called a singular integral operator of convolution type if its kernel is translation- invariant, that is, it is of the form K(t, s) = K˜(t − s). We call a singular integral opera- tor a Calderón-Zygmund operator if both the kernel K and the adjoint kernel K0 given by K0(t, s) := K(s, t) (t, s 2 R) satisfy the classical Lipschitz or second standard condition, which is the condition that there exists a constant d > 0 such that 1+d ∗ jt − sj 0 D [K] ∗ = jK(t s) − K(t s )j < ( ¥) D¥ : sup 0 d , , L(X,Y) ¥, t,s,s02R, js − s j 2js−s0j<jt−sj or, in the case of a translation-invariant kernel, 1+d ∗ jtj H [K] ∗ = jK(t − s) − K(t)j < ( ¥) H¥ : sup d L(X,Y) ¥. t,s2R, jsj 2jsj<jtj Coifman’s inequality, which appears in a general form in [16, Theorem II], says that for every Calderón-Zygmund operator T, every p 2 (0, ¥) and every Muckenhoupt weight w 2 A¥(R) there exists a constant C, depending on p and the A¥-constant of w, such that Z Z p p ¥ (1.1) jT f jYw dt ≤ C (M f ) w dt for every f 2 Lc (R; X), R R M being the classical Hardy-Littlewood maximal operator on R. In other words, all p weighted L -estimates for M, for p 2 (0, ¥) and w 2 A¥(R), are inherited by the op- erator T. Namely, by Muckenhoupt’s theorem and Coifman’s inequality, a Calderón- p Zygmund operator extrapolates to a bounded linear operator on Lw(R; X) for every p 2 (1, ¥) and every Muckenhoupt weight w 2 Ap(R). This result is best possible in the sense that there are Calderón-Zygmund operators which are not bounded on p Lw(R; X) (p 2 (1, ¥)) if the weight w does not belong to Ap(R). At the same time, if p the operator T is bounded on Lw(R; X) for some w 2 A¥(R) and some p 2 (1, ¥), then, by Lebesgue’s differentiation theorem, Coifman’s inequality (1.1) holds for all p f 2 Lw(R; X). Subsequently, variants of Coifman’s inequality for singular integral operators of convolution type and of nonconvolution type with less regular kernels attracted the attention during the last decades; see, in particular, the pioneering results from Kurtz & Wheeden [28], Rubio de Francia, Ruiz & Torrea [42], Alvarez & Pérez [2]. Following the terminology from [42, Definition 1.1, Part III], we say that a kernel K satisfies the WEIGHTED INEQUALITIES FOR SINGULAR INTEGRAL OPERATORS 3 condition (Dr) (r 2 [1, ¥)) if ¥ 1 1 m Z r 0 0 0 0 r (Dr) [K]D := sup js − s j r 2 r jK(t, s) − K(t, s )j dt < ¥, r ∑ 0 L(X,Y) s6=s0 m=1 Sm(s,s ) 0 m 0 0 m+1 0 where Sm(s, s ) := ft 2 R : 2 js − s j < jt − s j ≤ 2 js − s jg (m 2 N), and it satisfies the condition (D¥) if ¥ [ ] = j − 0j m j ( ) − ( 0)j < (D¥) K D¥ : sup s s ∑ 2 sup K t, s K t, s L(X,Y) ¥. 0 0 s6=s m=1 t2Sm(s,s ) 0 Moreover, we say that a kernel K satisfies the condition (Dr) for some r 2 [1, ¥] if 0 0 ( ) [ ] 0 = [ ] its adjoint kernel K satisfies the condition Dr , and then we set K Dr : K Dr . If a kernel K satisfies the condition (Dr) for some r 2 [1, ¥], then it satisfies also the condition (Dq) for every q 2 [1, r]. Finally, a translation-invariant kernel satisfies the 0 condition (Dr) if and only if it satisfies (Dr), which in the situation of a translation- invariant kernel we denote also by (Hr); see Lorente, Riveros & de la Torre [31], but also Kurtz & Wheeden [28]. It means that ¥ 1 1 m Z r r0 r0 r (Hr) [K]Hr := sup jsj 2 jK(t − s) − K(t)j dt < ¥ ∑ m m+1 L(X,Y) s6=0 m=1 2 jsj<jtj≤2 jsj if r 2 [1, ¥), and ¥ [ ] = j j m j ( − ) − ( )j < (H¥) K H¥ : sup s ∑ 2 sup K t s K t L(X,Y) ¥. s6=0 m=1 2mjsj<jtj≤2m+1jsj 1 Note that the condition (H¥) for a function K 2 Lloc(R n f0g; L(X)) is weaker than ∗ the second standard condition (H¥). If a kernel K satisfies the (Hr) condition, then the associated singular integral op- 0 erator T satisfies Coifman’s inequality (1.1) with M replaced by Mr0 (r 2 (1, ¥], r := r 1/r 1 r/(r − 1)), where Mr f := (M(j f j )) for every f 2 Lloc(R); see [42, Theorem 1.3, Part II] for the case r < ¥ and [2, Theorem 2.1] for the case r = ¥, and [42, Part III] for corresponding results in the nonconvolution case. By Martell, Pérez & Trujillo- González [32, Theorem 3.2], these variants of Coifman’s inequality are sharp in the 0 sense that the maximal operator Mr0 cannot be replaced by Ms for s < r .
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