Fourier multipliers for Hölder continuous functions and maximal regularity

Wolfgang Arendt; Charles Batty; Shangquan Bu

Displaying similar documents to “Fourier multipliers for Hölder continuous functions and maximal regularity”

Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces

Shangquan Bu, Yi Fang (2008)

Studia Mathematica

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We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $(P) u^{''} (t) + α u^{'} (t) + d / d t (\int_{- \infty}^{t} b (t - s) u (s) d s) = A u (t) - \int_{- \infty}^{t} a (t - s) A u (s) d s + f (t)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u’(0) = u’(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient...

Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet (2012)

Colloquium Mathematicae

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Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T \otimes I d_{X}$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^{p} (G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient...

Multilinear Fourier multipliers with minimal Sobolev regularity, I

Loukas Grafakos, Hanh Van Nguyen (2016)

Colloquium Mathematicae

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We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_{k}}$ , $0 < p_{k} \leq 1$ , to Lebesgue spaces $L^{p}$ . These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral...

Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

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If $x$ is a bounded function on $Z$ , the multiplier with symbol $x$ (denoted by $M_{x})$ is defined by $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . We give some conditions on $x$ ensuring the “interpolation inequality” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (here $1 < p < q$ and $α = α (p, q, x)$ is between 0 and 1). In most cases considered $M_{x}$ fails to have stronger $L^{1}$ -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E \subset Z$ every positive sequence in $ℓ^{2} (E)$ can be majorized by the sequence ${$ $| \hat{f} (n) |}_{n \in E}$ for some continuous funtion $f$ with spectrum...

Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem

Anthony Carbery (1988)

Annales de l'institut Fourier

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We show that the maximal operator associated to the family of rectangles in $R^{3}$ one of whose sides is parallel to $(1, 2^{j}, 2^{k})$ for some j,k $\in H Z$ is bounded on $L^{p}$ , $1 < p < \infty$ . We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

Rearrangement of coefficients of Fourier series on $S U_{2}$

Giancarlo Travaglini (1987)

Colloquium Mathematicae

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Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck (2001)

Studia Mathematica

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We consider the maximal regularity problem for the discrete time evolution equation $u_{n + 1} - T u ₙ = f ₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${(T ⁿ)}_{n \in ℕ ₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent...

Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier (2001)

Colloquium Mathematicae

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We study the space of functions φ: ℕ → ℂ such that there is a Hilbert space H, a power bounded operator T in B(H) and vectors ξ, η in H such that φ(n) = ⟨Tⁿξ,η⟩. This implies that the matrix ${(φ (i + j))}_{i, j \geq 0}$ is a Schur multiplier of B(ℓ₂) or equivalently is in the space (ℓ₁ ⊗̌ ℓ₁)*. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of H¹ which we call “shift-bounded”. We show that there is a φ which...

Well-posedness of second order degenerate differential equations in vector-valued function spaces

Shangquan Bu (2013)

Studia Mathematica

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Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces $L^{p} (, X)$ , periodic Besov spaces $B_{p, q}^{s} (, X)$ and periodic Triebel-Lizorkin spaces $F_{p, q}^{s} (, X)$ , where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M)....

The space of multipliers into $l_{1}$

P. Wojtaszczyk (1979)

Annales Polonici Mathematici

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A multiplier theorem for Fourier series in several variables

Nakhle Asmar, Florence Newberger, Saleem Watson (2006)

Colloquium Mathematicae

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We define a new type of multiplier operators on $L^{p} (^{N})$ , where $^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^{p} (^{N})$ , to which the theorem applies as a particular example.

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

Norm convergence of Fejér means of two-dimensional Walsh-Fourier series

Ushangi Goginava (2011)

Banach Center Publications

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The main aim of this paper is to prove that there exists a martingale $f \in H_{1 / 2}$ such that the Fejér means of the two-dimensional Walsh-Fourier series of f is not uniformly bounded in the space weak- $L_{1 / 2}$ .

A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)

Journal of the European Mathematical Society

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We strengthen the Carleson-Hunt theorem by proving $L^{p}$ estimates for the $r$ -variation of the partial sum operators for Fourier series and integrals, for $r > 𝚖𝚊𝚡 {p^{'}, 2}$ . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Maximal regularity for second order non-autonomous Cauchy problems

Charles J. K. Batty, Ralph Chill, Sachi Srivastava (2008)

Studia Mathematica

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We consider some non-autonomous second order Cauchy problems of the form ü + B(t)u̇ + A(t)u = f(t ∈ [0,T]), u(0) = u̇(0) = 0. We assume that the first order problem u̇ + B(t)u = f(t ∈ [0,T]), u(0) = 0, has $L^{p}$ -maximal regularity. Then we establish $L^{p}$ -maximal regularity of the second order problem in situations when the domains of B(t₁) and A(t₂) always coincide, or when A(t) = κB(t).

Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

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Writing $(T_{R}^{λ} f) \hat{} (ξ) = (1 - | ξ |^{2} / R^{2})_{+}^{λ} \hat{f} (ξ)$ . E. Stein conjectured $∥ {(\sum_{j} | T_{R_{j}}^{λ} f_{i} |^{2})}^{1 / 2} ∥_{p} \leq C ∥ {(\sum_{j} | f_{j} |^{2})}^{1 / 2} ∥_{p}$ for $λ > 0$ , $\frac{4}{3} \leq p \leq 4$ and $C = C_{λ, p}$ . We prove this conjecture. We prove also $f (x) = {lim}_{j \to \infty} T_{2^{j}}^{λ} f (x)$ a.e. We only assume $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1, \infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^{p}$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^{p}$ ...

Inductive extreme non-Arens regularity of the Fourier algebra A(G)

Zhiguo Hu (2002)

Studia Mathematica

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Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)*, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family ℋ of open subgroups of G such that: (1) for each H ∈ ℋ, A(H) is extremely non-Arens regular; (2) $V N (G) = ⋃_{H \in ℋ} V N (H)$ and $V N (G) / W A P (G ̂) = ⋃_{H \in ℋ} [V N (H) / W A P (H ̂)]$ ; (3) $A (G) = ⋃_{H \in ℋ} A (H)$ and it is a WAP-strong inductive union in the sense that the unions in...

The Herz-Schur multiplier norm of sets satisfying the Leinert condition

Éric Ricard, Ana-Maria Stan (2011)

Colloquium Mathematicae

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It is well known that in a free group , one has $| | χ_{E} {| |}_{M_{c b} A ()} \leq 2$ , where E is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert condition depends only on its cardinality. Consequently, based on a result of Wysoczański, we obtain a formula for $| | χ_{E} {| |}_{M_{c b} A ()}$ .

Multipliers of the space $A^{p}$

I. Jovanović (1986)

Matematički Vesnik

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Maximal operators of Fejér means of double Vilenkin-Fourier series

István Blahota, György Gát, Ushangi Goginava (2007)

Colloquium Mathematicae

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The main aim of this paper is to prove that the maximal operator $σ ₀ * : = s u p ₙ | σ_{n, n} |$ of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space $H_{1 / 2}$ to the space weak- $L_{1 / 2}$ .