Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems

?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      2. Calderón–Zygmund estimates for a class of obstacle problems with nonstandard growth      Jihoon Ok 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(4):50 We prove Calderón–Zygmund type estimates for obstacle problems with so-called \({L^{p(\cdot)} \log L}\)-growth. We also find suitable conditions on the variable exponent \({p(\cdot)}\) and the coefficients of the obstacle problems to obtain desired estimates.      3. Calderón–Zygmund estimates for parabolic measure data equations      Paolo Baroni  Jens Habermann 《Journal of Differential Equations》2012,252(1):412-447       4. Global Calderón–Zygmund theory for nonlinear parabolic systems      Verena Bögelein 《Calculus of Variations and Partial Differential Equations》2014,51(3-4):555-596 We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.      5. Nonlinear Calderón–Zygmund inequalities for maps      Batu Güneysu  Stefano Pigola 《Annals of Global Analysis and Geometry》2018,54(3):353-364 Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.      6. Global Calderón–Zygmund estimates for a class of nonlinear elliptic equations with Neumann data      Fengping Yao 《Journal of Mathematical Analysis and Applications》2018,457(1):551-567 In this paper we obtain the following global Calderón–Zygmund estimates $B\left(\left|\mathbf{f}\right|\right)\in L^q(\mathrm{\Omega })?B\left(\left|\mathrm{?}u\right|\right)\in L^q(\mathrm{\Omega })\text{for any}q\ge 1$ in a convex domain Ω of the weak solution for a class of quasilinear elliptic equations with the Neumann data $\begin{array}{ccc}\hfill \text{div}\left(a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u\right)& \hfill =\hfill & \text{div}\left(a\left(\left|\mathbf{f}\right|\right)\mathbf{f}\right)\text{in}\mathrm{\Omega },\hfill \\ \hfill a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u?\nu & \hfill =\hfill & a\left(\left|\mathbf{f}\right|\right)\mathbf{f}?\nu \text{on}?\mathrm{\Omega },\hfill \end{array}$ where $B(t)={\int }_0^t\tau a(\tau )d\tau$ for $t\ge 0$. Moreover, we remark that $B(t)=t^p\log ?(1+t)$ satisfies the given conditions in this work.      7. Rellich and Calderón–Zygmund inequalities for an operator with discontinuous coefficients      G. Metafune  M. Sobajima  C. Spina 《Annali di Matematica Pura ed Applicata》2016,195(4):1305-1331       8. Weighted Calderón–Zygmund Decomposition with Some Applications to Interpolation      D. V. Rutsky 《Journal of Mathematical Sciences》2015,209(5):783-791       9. An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators      Stockdale  Cody B.  Wick  Brett D. 《Journal of Fourier Analysis and Applications》2019,25(5):2635-2652 Journal of Fourier Analysis and Applications - The purpose of this article is to provide an alternative proof of the weak-type $$\left( 1,\ldots ,1;\frac{1}{m}\right) $$ estimate for m-multilinear...      10. Calderón–Zygmund operators in the Bessel setting      Jorge J. Betancor  Alejandro J. Castro  Adam Nowak 《Monatshefte für Mathematik》2012,167(3-4):375-403 We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.      11. Calderón–Zygmund Operators on Weighted Hardy Spaces      Ming-Yi Lee 《Potential Analysis》2013,38(3):699-709 We study the boundedness of Calderón–Zygmund operators on weighted Hardy spaces $H^p_w$ using Littlewood-Paley theory. It is shown that if a Calderón–Zygmund operator T satisfies T *1?=?0, then T is bounded on $H^p_w$ for $w\in A_{p(1+\frac\varepsilon n)}$ and $\frac n{n+\varepsilon}<p\le1$ , where ε is the regular exponent of the kernel of T.      12. Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators          下载免费PDF全文 Yan Lin  Guo Ming Zhang 《数学学报(英文版)》2016,32(11):1297-1311 In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.      13. Weighted Calderón–Zygmund and Rellich inequalities in L^p      Giorgio Metafune  Motohiro Sobajima  Chiara Spina 《Mathematische Annalen》2015,361(1-2):313-366       14. Hölder–Zygmund estimates for degenerate parabolic systems      Sebastian Schwarzacher 《Journal of Differential Equations》2014       15. Reaction–diffusion systems with p(x)-growth for image denoising      Zhichang Guo  Qiang Liu  Jiebao Sun  Boying Wu 《Nonlinear Analysis: Real World Applications》2011,12(5):2904-2918       16. Sharp Weighted Bounds for Multilinear Maximal Functions and Calderón–Zygmund Operators      Wendolín Damián  Andrei K. Lerner  Carlos Pérez 《Journal of Fourier Analysis and Applications》2015,21(1):161-181       17. Calderón–Zygmund kernels and rectifiability in the plane      V. Chousionis  J. Mateu  L. Prat  X. Tolsa 《Advances in Mathematics》2012,231(1):535-568       18. Generalized Calderón–Zygmund operators on homogeneous groups and applications      Der-Chen Chang  Ming-Yi Lee 《Applicable analysis》2013,92(5):531-554       19. The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators      Kangwei?Li  Kabe?MoenEmail author  Wenchang?Sun 《Journal of Fourier Analysis and Applications》2014,20(4):751-765 We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).      20. Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices   总被引：1，自引：0，他引：1   Alejandro J. Castro  Tomasz Z. Szarek 《数学学报(英文版)》2014,30(4):637-648 We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.

Calderón–Zygmund estimates for a class of obstacle problems with nonstandard growth

We prove Calderón–Zygmund type estimates for obstacle problems with so-called \({L^{p(\cdot)} \log L}\)-growth. We also find suitable conditions on the variable exponent \({p(\cdot)}\) and the coefficients of the obstacle problems to obtain desired estimates.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

Nonlinear Calderón–Zygmund inequalities for maps

Being motivated by the problem of deducing \(\mathsf {L}^{p}\)-bounds on the second fundamental form of an isometric immersion from \(\mathsf {L}^{p}\)-bounds on its mean curvature vector field, we prove a nonlinear Calderón–Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.     

Global Calderón–Zygmund estimates for a class of nonlinear elliptic equations with Neumann data

In this paper we obtain the following global Calderón–Zygmund estimates

$B\left(\left|\mathbf{f}\right|\right)\in L^q(\mathrm{\Omega })?B\left(\left|\mathrm{?}u\right|\right)\in L^q(\mathrm{\Omega })\text{for any}q\ge 1$

in a convex domain Ω of the weak solution for a class of quasilinear elliptic equations with the Neumann data

$\begin{array}{ccc}\hfill \text{div}\left(a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u\right)& \hfill =\hfill & \text{div}\left(a\left(\left|\mathbf{f}\right|\right)\mathbf{f}\right)\text{in}\mathrm{\Omega },\hfill \\ \hfill a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u?\nu & \hfill =\hfill & a\left(\left|\mathbf{f}\right|\right)\mathbf{f}?\nu \text{on}?\mathrm{\Omega },\hfill \end{array}$

where $B(t)={\int }_0^t\tau a(\tau )d\tau$ for $t\ge 0$. Moreover, we remark that

$B(t)=t^p\log ?(1+t)$

satisfies the given conditions in this work.     

An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

Journal of Fourier Analysis and Applications - The purpose of this article is to provide an alternative proof of the weak-type $$\left( 1,\ldots ,1;\frac{1}{m}\right) $$ estimate for m-multilinear...     

Calderón–Zygmund operators in the Bessel setting

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.     

Calderón–Zygmund Operators on Weighted Hardy Spaces

We study the boundedness of Calderón–Zygmund operators on weighted Hardy spaces $H^p_w$ using Littlewood-Paley theory. It is shown that if a Calderón–Zygmund operator T satisfies T ^*1?=?0, then T is bounded on $H^p_w$ for $w\in A_{p(1+\frac\varepsilon n)}$ and $\frac n{n+\varepsilon}<p\le1$ , where ε is the regular exponent of the kernel of T.     

Weighted Estimates for Commutators of Strongly Singular Calderón–Zygmund Operators

In this paper,the authors establish the boundedness of commutators generated by strongly singular Calderón–Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces.Moreover,the corresponding results for commutators generated by strongly singular Calderón–Zygmund operators and weighted Lipschitz functions can also be obtained.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.