Littlewood-Paley Theorem Research Articles

Littlewood-Paley Theorem, Nikolskii Inequality, Besov Spaces, Fourier and Spectral Multipliers on Graded Lie Groups

In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.

2D Linear Canonical Transforms on Lp and Applications

As Fourier transformations of Lp functions are the mathematical basis of various applications, it is necessary to develop Lp theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this Lp theory for 1≤p<∞. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L1(R2). Together with the Plancherel identity and Hausdorff–Young inequality, we establish Lp(R2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L1(R2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images.

Modulation Spaces, Multipliers Associated with the Special Affine Fourier Transform

We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space $${{\varvec{M}}}^{r,s}_A$$ in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.

Fractional Fourier transforms on L and applications

This paper is devoted to the L p ( R ) theory of the fractional Fourier transform (FRFT) for 1 ≤ p < 2 . In view of the special structure of the FRFT, we study FRFT properties of L 1 functions, via the introduction of a suitable chirp operator. However, in the L 1 ( R ) setting, problems of convergence arise even when basic manipulations of functions are performed. We overcome such issues and study the FRFT inversion problem via approximation by suitable means, such as the fractional Gauss and Abel means. We also obtain the regularity of fractional convolution and results on pointwise convergence of FRFT means. Finally we discuss L p multiplier results and a Littlewood-Paley theorem associated with FRFT.

О существовании граничных и начальных значений для вырождающихся параболических уравнений в областях с ляпуновской границей

В данной работе, являющейся продолжением [1], устанавливаются необходимые и достаточные условия того, чтобы решение параболического уравнения 2-го порядка с боковой границей, принадлежащей классу $C^{1+\lambda}$, $\lambda &gt;0$, вырождающегося на границе области, имело предел в среднем на боковой поверхности цилиндрической области и предел в среднем на ее нижнем основании, и исследуется вопрос об однозначной разрешимости первой смешанной задачи для такого уравнения в случае, когда граничная и начальная функции принадлежат пространствам типа $L_2$. Наиболее близкими к рассматриваемому кругу вопросов являются теоремы Ф. Рисса и Ж. Литтлвуда и Р. Пэли, в которых даются критерии предельных значений в $L_p$, $p &gt; 1$, аналитических в единичном круге функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина [2–4]. Условие гладкости границы ($\partial Q \in C^2$) можно ослабить (см. [5]). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерии существования граничного значения установлены в работах [4, 6–8]. При этом, как показано в работе [7], все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса [9] и Литтлвуда — Пэли [10, 11]. This work, being a continuation of [1], establishes the necessary and sufficient conditions for the solution of the second-order parabolic equations with a lateral boundary from the class $C^{1+\lambda}$, $\lambda &gt;0$, degenerating on the boundary of the domain, to have an average limit on the lateral surface of the cylindrical domain and the limit in the mean on its lower base. Also, we study the question of the unique solvability of the first mixed problem for such equations in the case when the boundary and initial functions belong to spaces of the типа $L_2$ type. The closest to the questions under consideration are the theorems of F. Riesz and J. Littlewood and R. Paley, in which criteria are given for the limit values in $L_p$, $p &gt; 1$, of the functions analytic in the unit disk. Further development of this topic for uniformly elliptic equations was obtained in the papers by V. P. Mikhailov and A. K. Gushchin [2–4]. The boundary smoothness condition ($\partial Q \in C^2$) can be weakened (see [5]). Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criteria for the existence of a boundary value were established in [4, 6–8]. In this case, as shown in [7], all directions of the acceptance of boundary values for uniformly elliptic equations turn out to be equal, while the solution has a property similar to the property of continuity with respect to the set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation is more complicated. In this case, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are different from zero (the Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equations are very close to the properties of uniformly elliptic equations; in particular, in this situation analogues of the Riesz [9] and Littlewood–Paley theorems [10, 11] are valid.

Multipliers for Besov spaces on graded Lie groups

In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.

An analogue of the Littlewood–Paley theorem for orthoprojectors onto wavelet subspaces

We prove an analogue of the Littlewood–Paley theorem for orthoprojectors onto wavelet subspaces corresponding to a non-isotropic multiresolution analysis generated by the tensor product of smooth scaling functions of one variable with sufficiently rapid decay at infinity.

On weighted 𝐿² estimates for solutions of the wave equation

In this paper we consider weighted L 2 L^2 integrability for solutions of the wave equation. For this, we obtain some weighed L 2 L^2 estimates for the solutions with weights in Morrey-Campanato classes. Our method is based on a combination of bilinear interpolation and a localization argument which makes use of the Littlewood-Paley theorem and a property of Hardy-Littlewood maximal functions. We also apply the estimates to the problem of well-posedness for wave equations with potentials.

An analogue of the Littlewood-Paley theorem for orthoprojectors onto wavelet subspaces

Доказано утверждение, представляющее собой аналог теоремы Литтлвуда-Пэли для ортопроекторов на подпространства всплесков, соответствующие неизотропному кратномасштабному анализу, порожденному тензорным произведением гладких достаточно быстро стремящихся к нулю на бесконечности масштабирующих функций одной переменной. Библиография: 20 наименований.

A Littlewood–Paley type theorem and a corollary

We prove an analogue of the Littlewood–Paley theorem for orthoprojectors onto mutually orthogonal subspaces of piecewise-polynomial functions on the cube . This yields upper bounds for the norms of functions in in terms of the corresponding norms of the projections to subspaces of piecewise-polynomial functions of several variables. We use these results to obtain upper bounds for the Kolmogorov widths of Besov classes of (non-periodic) functions satisfying mixed Hölder conditions.

Littlewood-paley theorem and multipliers on the quantum torus

The Littlewood-Paley and Marcinkiewicz's multiplier theorems on the quan-tum torus are established. An key ingredient of the proof is vector-valued Littlewood-Paley and noncommutative Khinchin's inequalities.

Littlewood-Paley theorem for arbitrary intervals: Weighted estimates

Let 1 < r < 2 and let b is a weight on ℝ such that $b^{ - \frac{1} {{r - 1}}} $ satisfies the Muckenhoupt condition Ar′/2 (r′ is the exponent conjugate to r). If fj are functions whose Fourier transforms are supported on mutually disjoint intervals, then $$\left\| {\sum\nolimits_j {fj} } \right\|_{L^p \left( {\mathbb{R},{\kern 1pt} b} \right)} \leqslant C\left\| {\left( {\sum\nolimits_j {\left| {fj} \right|^2 } } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\|_{L^P \left( {\mathbb{R},{\kern 1pt} b} \right)} $$ for 0 < p ≤ r. Bibliography: 9 titles.

Littlewood–Paley theorem on spaces L p(t)(ℝ n )

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L p(t) (ℝ n ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ n , if and only if p(t) = const.

MULTIPLIER THEOREMS IN WEIGHTED SMIRNOV SPACES

The analogues of Marcinkiewicz multiplier theorem and Littlewood-Paley theorem are proved for p-Faber series in weighted Smirnov spaces defined on bounded and unbounded components of a rectifiable Jordan curve.

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

AbstractSeveral rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood–Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of ‘off-diagonal’ estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.

Littlewood-Paley theorem for Schrödinger operators

Let H be a Schrodinger operator on ℝ n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

On the Littlewood-Paley theorem for arbitrary intervals

We extend the results of Rubio de Francia and Bourgain by showing that, for arbitrary mutually disjoint intervals Δk ⊂ ℤ+, arbitrary p ∈, (0, 2], and arbitrary trigonometric polynomials f k with supp \(\hat f_k \subset \Delta _k \), we have $$\left\| {\sum\limits_k {f_k } } \right\|_{H^p (\mathbb{T})} \leqslant a_p \left\| {\left( {\sum\limits_k {\left| {f_k } \right|} ^2 } \right)^{1/2} } \right\|_{L^p (\mathbb{T})} $$ . The method is a development of that by Rubio de Francia. Bibliography: 9 titles.

Lp([0,1])-characterizations of multi-knot piecewise linear spectral sequences*

Abstract There exists a class of new orthonormal basis for L2([0,1]), whose exponential parts are multi-knot piecewise linear functions called spectral sequences. In this paper, we show that these bases constitute bases, but not unconditional bases, for L p ([0,1])with 1 < p < ∞, p≠2. In addition, we give the corresponding convergence theorem in L p , Carleson-Hunt theorem on almost everywhere convergence, Littlewood-Paley theorem and Poisson summation formula related to these bases.. * Supported by National Natural Science Foundation of China (Grant No. 10371122) ad the President Foundation of Graduate School of the Chinese Academy of Sciences (Grant No. yzjj200505)

Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth

AbstractWe introduce a Littlewood–Paley decomposition related to any sub‐Laplacian on a Lie group G of polynomial volume growth; this allows us to prove a Littlewood–Paley theorem in this general setting and to provide a dyadic characterization of Besov spaces B s,q p (G ), s ∈ ℝ, equivalent to the classical definition through the heat kernel. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

INHOMOGENEOUS CALDERÓN REPRODUCING FORMULAE ASSOCIATED TO PARA-ACCRETIVE FUNCTIONS ON METRIC MEASURE SPACES

Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type which includes metric measure spaces and some fractals, namely, $X$ is a set, $\rho$ is a quasi-metric on $X$ satisfying that there exist constants $C_0 \gt 0$ and $\theta \in (0,1]$ such that for all $x,\ x',\ y \in X$, $$ |\rho(x,y)-\rho(x',y)| \le C_0 \rho(x,x')^\theta [\rho(x,y) + \rho(x',y)]^{1-\theta}, $$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d \gt 0$, all $x \in X$ and all $0 \lt r \lt \operatorname{diam} X$, \[ \mu(\{ y \in X: \rho(x,y) \lt r \}) \sim r^{d}. \] In this paper, we first obtain the boundedness of Calderón-Zygmund operators on spaces of test functions; and using this, we then establish the continuous Calderón reproducing formulae associated with a given para-accretive function, which is a key tool for developing the theory of Besov and Triebel-Lizorkin spaces associated with para-accretive functions. By the Calderón reproducing formulae, we finally obtain a Littlewood-Paley theorem on the inhomogeneous $g$-function which gives a new characterization of Lebesgue spaces $L^{p}(X)$ for $p \in (1,\infty)$ and generalizers a corresponding result of David, Journé and Semmes.