Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces

Recently, F. Nazarov, S. Treil and A. Volberg (and independently X. Tolsa) have extended the classical theory of Calderón-Zygmund operators to the context of a "non-homogeneous" space (X, d, µ), where, in particular, the measure µ may be non-doubling. In the present work we study weighted inequalities for these operators. Specifically, for 1 < p < [infinity], we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted Lp inequality holds. We deal with this problem by developing a vector-valued theory for Calderón-Zygmund operators on non-homogeneous spaces which is interesting in its own right. For the case of the Cauchy integral operator, which is the most important example, we even prove that the conditions for the weights are also necessary

Factorization of operators through subspaces of L-1-spaces

[EN] We analyze domination properties and factorization of operators in Banach spaces through subspaces of L1-spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of L1-spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.Research supported by MINECO/FEDER under projects MTM2014-53009-P (J.M Calabuig), MTM2014-54182-P (J. Rodriguez) and MTM2012-36740-C02-02 (E. A. Sanchez-Perez).Calabuig, JM.; Rodríguez, J.; Sánchez Pérez, EA. (2017). Factorization of operators through subspaces of L-1-spaces. Journal of the Australian Mathematical Society. 103(3):313-328. https://doi.org/10.1017/S1446788716000513S3133281033Lindenstrauss, J., & Tzafriri, L. (1979). Classical Banach Spaces II. doi:10.1007/978-3-662-35347-9Pisier, G. (1986). Factorization of Linear Operators and Geometry of Banach Spaces. CBMS Regional Conference Series in Mathematics. doi:10.1090/cbms/060Okada, S., Ricker, W. J., & Sánchez Pérez, E. A. (2008). Optimal Domain and Integral Extension of Operators. doi:10.1007/978-3-7643-8648-1Lacey, H. E. (1974). The Isometric Theory of Classical Banach Spaces. doi:10.1007/978-3-642-65762-7Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., & Sánchez-Pérez, E. A. (2005). Vector measure Maurey–Rosenthal-type factorizations and ℓ-sums of L1-spaces. Journal of Functional Analysis, 220(2), 460-485. doi:10.1016/j.jfa.2004.06.010Juan, M. A., & Sánchez Pérez, E. A. (2013). Maurey-Rosenthal domination for abstract Banach lattices. Journal of Inequalities and Applications, 2013(1). doi:10.1186/1029-242x-2013-213Avilés, A., Cabello Sánchez, F., Castillo, J. M. F., González, M., & Moreno, Y. (2013). On separably injective Banach spaces. Advances in Mathematics, 234, 192-216. doi:10.1016/j.aim.2012.10.013Defant, A., & Sánchez Pérez, E. A. (2004). Maurey–Rosenthal factorization of positive operators and convexity. Journal of Mathematical Analysis and Applications, 297(2), 771-790. doi:10.1016/j.jmaa.2004.04.047DEFANT, A., & PÉREZ, E. A. S. (2009). Domination of operators on function spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 146(1), 57-66. doi:10.1017/s0305004108001734Bartle, R. G., Dunford, N., & Schwartz, J. (1955). Weak Compactness and Vector Measures. Canadian Journal of Mathematics, 7, 289-305. doi:10.4153/cjm-1955-032-1Rosenthal, H. P. (1974). A Characterization of Banach Spaces Containing l1. Proceedings of the National Academy of Sciences, 71(6), 2411-2413. doi:10.1073/pnas.71.6.2411Diestel, J., Jarchow, H., & Tonge, A. (1995). Absolutely Summing Operators. doi:10.1017/cbo9780511526138Rueda, P., & Sánchez-Pérez, E. A. (2015). Compactness in spaces of p-integrable functions with respect to a vector measure. Topological Methods in Nonlinear Analysis, 45(2), 641. doi:10.12775/tmna.2015.030Rosenthal, H. P. (1973). On Subspaces of L p. The Annals of Mathematics, 97(2), 344. doi:10.2307/1970850Diestel, J., & Uhl, J. (1977). Vector Measures. Mathematical Surveys and Monographs. doi:10.1090/surv/015[16] M. Mastyło and E. A. Sánchez-Pérez , ‘Factorization of operators through Orlicz spaces’, Bull. Malays. Math. Sci. Soc. doi:10.1007/s40840-015-0158-5, to appear.Calabuig, J. M., Lajara, S., Rodríguez, J., & Sánchez-Pérez, E. A. (2014). Compactness in L1of a vector measure. Studia Mathematica, 225(3), 259-282. doi:10.4064/sm225-3-6Defant, A. (2001). Positivity, 5(2), 153-175. doi:10.1023/a:1011466509838Fabian, M., Habala, P., Hájek, P., Montesinos, V., & Zizler, V. (2011). Banach Space Theory. CMS Books in Mathematics. doi:10.1007/978-1-4419-7515-

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

The ℓs\ell^s-boundedness of a family of integral operators on UMD Banach function spaces

We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.Comment: 13 pages. Generalization of arXiv:1410.665

Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let \phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz) be a function such that $\phi(x,\cdot)$ is an Orlicz function and $\phi(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_{\phi}(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{\phi}(\mathbb{R}^n)$ which is further proved to be the dual space of $h_{\phi}(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipliers for the local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^{\phi}(\mathbb{R}^n)$, characterized by E. Nakai and K. Yabuta, is just the dual of L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n), where $\phi$ is an increasing function on $(0,\infty)$ satisfying some additional growth conditions and $\Phi_0$ a Musielak-Orlicz function induced by $\phi$. Characterizations of $h_{\phi}(\mathbb{R}^n)$, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_{\phi}(\mathbb{R}^n)$, from which, the authors further deduce some criterions for the boundedness on $h_{\phi}(\mathbb{R}^n)$ of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on $h_{\phi}(\mathbb{R}^n)$.Comment: Sci. China Math. (to appear

Anisotropic Singular Integrals in Product Spaces

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by {\mathcal A}_\infty(\rnm; \vec A) the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.Comment: Sci. China Math., to appea