Multipliers of double Fourier–Haar series
Tóm tắt
The multipliers of double Fourier–Haar series for functions from anisotropic Lorentz spaces are investigated. Necessary and sufficient conditions are obtained for the sequence
$$\lambda =\{\lambda _{k_1k_2}^{j_1j_2}\}$$
to belong to the class
$$m(L_{\bar{p},\bar{r}}\rightarrow L_{\bar{q},\bar{s}})$$
. In particular, the case is described when
$$\bar{s}<\bar{r}$$
, which is a new result in the one-dimensional case as well.
Tài liệu tham khảo
Akylzhanov, R., Nursultanov, E., Ruzhansky, M.: Hardy–Littlewood–Paley inequalities and Fourier multipliers on SU(2). Stud. Math. 234(1), 1–29 (2016)
Akylzhanov, R., Ruzhansky, M.: Net spaces on lattices, Hardy–Littlewood type unequalities, and their converses. Eur. Math. J. 8(3), 10–27 (2017)
Akylzhanov, R., Ruzhansky, M., Nursultanov, E.: Hardy–Littlewood, Hausdorff–Young–Paley inequalities, and \(L_p\rightarrow L_q\) Fourier multipliers on compact homogeneous manifolds. J. Math. Anal. Appl. 479(2), 1519–1548 (2019)
Bashirova, A.N., Kalidolday, A.H., Nursultanov, E.D.: Interpolation theorem for anisotropic net spaces. arXiv:2009.00609 (2020)
Bashirova, A.N., Nursultanov, E.D.: The Hardy–Littlewood theorem for double Fourier–Haar series from Lebesgue spaces \(L_{\bar{p}}[0,1]\) with mixed metric and from net spaces \(N_{\bar{p}, \bar{q}}(M)\). arXiv:2009.01105 (2020)
Bekmaganbetov, K.A.: Interpolation theorem for \(l^\sigma _q(L_{p\tau })\)\(L_{p\tau }(l^\sigma _q)\) spaces. Bulletin of the Kazakh National University. Series mathematics, mechanics, computer science. 56(1), 30–42 (2008)
Bekmaganbetov, K.A., Nursultanov, E.D.: Embedding theorems for anisotropic Besov spaces \(B_{\bar{p},\bar{r}}^{\bar{\alpha },\bar{q}}\left([0,2\pi )^n\right)\). Izv. Math. 73(4), 655–668 (2009)
Bryskin, I.B., Lelond, O.V., Semenov, E.M.: Multipliers of the Fourier–Haar series. Sib. Math. J. 41(4), 626–633 (2000)
Burkholder, D.L.: A nonlinear partial differential equation and unconditional constant of the Haar system in \(L_p\). Bull. Am. Math. Soc. 7, 591–595 (1982)
Fernandez, D.L.: Lorentz spaces with mixed norms. J. Funct. Anal. 25(2), 128–146 (1977)
Fernandez, D.L.: Interpolation of \(2^n\) Banach spaces. Stud. Math. (PRL) 65(2), 175–201 (1979)
Fernandez, D.L.: Interpolation of \(2^n\) Banach space and the Calderon spaces. Proc. Lond. Math. Soc. 5(6), 143–162 (1988)
Girardi, M.: Operator-valued Fourier Haar multipliers. J. Math. Anal. Appl. 325, 1314–1326 (2007)
Kashin, B.S., Saakyan, A.A.: Orthogonal Series, p. 496. Nauka, Moscow (1984)
Lelond, O.V., Semenov, E.M., Uksusov, S.N.: The space of Fourier–Haar multipliers. Sib. Math. J. 46(1), 103–110 (2005)
Lizorkin, P.I.: Multipliers of Fourier integrals in the spaces \(L_{p,\theta }\) (Russian). Trudy Mat. Inst. Steklov 89, 231–248 (1967)
Lizorkin, P.I.: On the theory of Fourier multipliers. Proc. Steklov Inst. Math. 173, 161–176 (1987)
Marcinkiewicz, J., Zygmund, A.: Some theorems on orthogonal systems. Fund. Math. 28, 309–335 (1937)
Novikov, I., Semenov, E.: Haar Series and Linear Operators, p. 218. Cluver Acad. Publ., Dordrecht (1997)
Nursultanov, E.D.: Concerning the multiplicators of Fourier series in the trigonometric system. Math. Notes 63(2), 205–214 (1998)
Nursultanov, E.D.: Net spaces and inequalities of Hardy–Littlewood type. Sb. Math. 189(3), 399–419 (1998)
Nursultanov, E.D., Tleukhanova, N.T.: Multipliers of multiple Fourier series. Proc. Steklov Inst. Math. 227, 231–236 (1999)
Nursultanov, E.D., Tleukhanova, N.T.: Lower and upper bounds for the norm of multipliers of multiple trigonometric Fourier series in Lebesgue spaces. Funct. Anal. Appl. 34(2), 151–153 (2000)
Nursultanov, E.D.: On the coefficients of multiple Fourier series in \(L_p\)-spaces. Izv. Math. 64(1), 93–120 (2000)
Nursultanov, E.D., Aubakirov, T.U.: The Hardy–Littlewood theorem for Fourier–Haar series. Math. Notes 73(3), 314–320 (2003)
Nursultanov, E.D.: Interpolation theorems for anisotropic function spaces and their applications. Rep. Russ. Acad. Sci. 394(1), 1–4 (2004)
Nursultanov, E.D.: Nikol’skii’s Inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space. Proc. Steklov Inst. Math. 255, 185–202 (2006)
Nursultanov, E., Tikhonov, S.: Net spaces and boundedness of integral operators. J. Geom. Anal. 21, 950–981 (2011)
Nursultanov, E., Sarybekova, L., Tleukhanova, N.: Some new Fourier multiplier results of Lizorkin and Hormander types. In: Functional Analysis in Interdisciplinary Applications, Springer Proc. Math. Stat., vol. 216, pp. 58–82. Springer, Cham (2017)
Persson, L.-E., Sarybekova, L., Tleukhanova, N.: A Lizorkin theorem on Fourier series multipliers for strong regular systems. In: Analysis for Science, Engineering and Beyond, Springer Proc. Math., vol. 6, pp. 305–317. Springer, Heidelberg (2012)
Sarybekova, L.O., Tararykova, T.V., Tleukhanova, N.T.: On a generalization of the Lizorkin theorem on Fourier multipliers. Math. Inequal. Appl. 13(3), 613–624 (2010)
Semenov, E.M., Uksusov, S.N.: Multipliers of the Haar series. Sib. Math. J. 53(2), 310–315 (2012)
Wark, H.M.: Operator-valued Fourier Haar multipliers on vector-valued \(L_1\) spaces. J. Math. Anal. Appl. 450, 1148–1156 (2017)
Yano, S.: On a lemma of Marcinkiewicz and its applications to Fourier series. Tohoku Math. J. 11, 195–215 (1959)
Yudin, V.A.: Spherical sums of Fourier series in \(L_p\). Math. Notes 46(2), 675–680 (1989)