In this paper, we have established:

First, if a subadditive operator T is weakly bounded in  and in  then there exists a  such that T is strongly bounded for any  for which , whenever

Second, assuming that the kernel of the integral operator T satisfies the regularity conditions and that we obtain

for all integrable functions, where the constant  depends only on,

Third, for a locally integrable function f and weight w belonging to  

harmonic analysis, singular integrals, convex semi-norms, interpolation theorem, Calderon-Zygmund decomposition, maximal operator.

Received: May 26, 2020; Accepted: July 3, 2020: Published: December 28, 2020

How to cite this article: Mykola Ivanovich Yaremenko, Application of Harmonic Analysis Methods to Singular Integrals and Convex Semi-Norms, Interpolation Theorems, International J. Functional Analysis, Operator Theory and Applications 13(1) (2021), 1-27. DOI: 10.17654/FA013010001

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285-320.
[2] K. Adimurthi and N. C. Phuc, Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations, Calc. Var. Partial Differential Equations 54(3) (2015), 3107-3139.
[3] A. Beni, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multiplier for modulation spaces, J. Funct. Anal. 246 (2007), 366-384.
[4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. 130(1) (1989), 189-213.
[5] L. A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51(1) (1998), 1-21.
[6] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.
[7] T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations 248(8) (2010), 1972-2002.
[8] C. Ortiz-Caraballo, C. Perez and E. Rela, Exponential decay estimates for singular integral operators, Math. Ann. 357 (2018), 1217-1243.
[9] A. G. Polimeridis, F. Vipiana, J. R. Mosig and D. R. Wilton, DIRECTFN: Fully numerical algorithms for high precision computation of singular integrals in Galerkin SIE methods, IEEE Trans. Antennas Propag. 61(6) (2013), 3112-3122.
[10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J., 1970.
[11] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations I, Hackensack, NJ, World Scientfic, 2011.
[12] F. Y. Wang, Distribution dependent SDEs for Landau type equations, Stochastic Processes and their Applications 128(2) (2018), 595-621.
[13] P. Xia, L. Xie, X. Zhang and G. Zhao Lq(Lp)-theory of stochastic differential equations, Stochastic Processes and their Applications 130(8) (2020), 5188-5211.
[14] Q. S. Zhang, Gaussian bounds for the fundamental solutions of  Manuscript Mathematica 93(1) (1997), 381-390.
[15] X. Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electronic Journal of Probability 16 (2011), 1096-1116.
[16] X. Zhang and G. Zhao, Singular Brownian diffusion processes, Communications in Mathematics and Statistics 6(4) (2018), 533-581.
[17] X. Zhang and G. Zhao, Stochastic Lagrangian path for Leray’s solutions of 3D Navier-Stokes equations, Commun. Math. Phys. (2020), 25pp. https://doi.org/10.1007/s00220-020-03888-w.
[18] Qiaoling Wang and Changyu Xia, Sharp bounds for the first non-zero Stekloff eigenvalues, J. Funct. Anal. 257(8) (2009), 2635-2644.
[19] Changwei Xiong, Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds, J. Funct. Anal. 275(12) (2018), 3245-3258.