Maximal regularity for second order non-autonomous Cauchy problems

Charles J. K. Batty; Ralph Chill; Sachi Srivastava

Displaying similar documents to “Maximal regularity for second order non-autonomous Cauchy problems”

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...

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 $L^{p}$ ...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let $S \subset L_{p}$ be a subspace of an $L_{p}$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S (Y) \subset L_{p} (Y)$ . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{- t B}$ is a positive contraction...

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

$L^{p}$ Maximal Regularity for Second Order Cauchy Problems is Independent of $p$

Ralph Chill, Sachi Srivastava (2008)

Bollettino dell'Unione Matematica Italiana

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If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^{p}$ maximal regularity for some $p\in(1,\infty)$ , then it has $L^{p}$ maximal regularity for every $p\in(1,\infty)$ .

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity,...

Evolution equations governed by Lipschitz continuous non-autonomous forms

Ahmed Sani, Hafida Laasri (2015)

Czechoslovak Mathematical Journal

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We prove $L^{2}$ $\dot{u} (t) + A (t) u (t) = f (t) for a.e. t \in [0, T], u (0) = u_{0},$ where the operator $A (t)$ arises from a time depending sesquilinear form $𝔞 (t, \cdot, \cdot)$ on a Hilbert space $H$ with constant domain $V .$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed...

A note on rare maximal functions

Paul Alton Hagelstein (2003)

Colloquium Mathematicae

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A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

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Let $M_{S}$ denote the strong maximal operator. Let $M_{x}$ and $M_{y}$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that $\int_{Q} M_{y} M_{x} h < \infty$ but $\int_{Q} M_{x} M_{y} h = \infty$ . It is shown that if f is a function supported on Q such that $\int_{Q} M_{y} M_{x} f < \infty$ but $\int_{Q} M_{x} M_{y} f = \infty$ , then there exists a set A of finite measure in ℝ² such that $\int_{A} M_{S} f = \infty$ .

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p, q}$ , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

Radial maximal function characterizations for Hardy spaces on RD-spaces

Loukas Grafakos, Liguang Liu, Dachun Yang (2009)

Bulletin de la Société Mathématique de France

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An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$ , there exists a $p_{0} \in (n / (n + 1), 1)$ such that for certain classes of distributions, the $L^{p} (𝒳)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p \in (p_{0}, \infty]$ . This result yields a radial maximal function characterization for Hardy spaces on $𝒳$ . ...

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

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The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_{p} (ℝ^{d})$ (in the case $p > 1$ ), but (in the case when $1 / p (\cdot)$ is log-Hölder continuous and $p_{-} = inf {p (x) : x \in ℝ^{d}} > 1$ ) on the variable Lebesgue spaces $L_{p (\cdot)} (ℝ^{d})$ , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1, 1)$ . In the present note we generalize Besicovitch’s covering theorem for the so-called $γ$ -rectangles. We introduce a general maximal operator $M_{s}^{γ, δ}$ and with the help of generalized $Φ$ -functions, the strong-...

Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows

Jonas Sauer (2016)

Czechoslovak Mathematical Journal

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We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal $L^{p}$ -regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G : = ℝ^{n - 1} \times ℝ / L ℤ$ to obtain an $ℛ$ -bound...

Maximal non-pseudovaluation subrings of an integral domain

Rahul Kumar (2024)

Czechoslovak Mathematical Journal

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The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let $R \subset S$ be an extension of domains. Then $R$ is called a maximal non-pseudovaluation subring of $S$ if $R$ is not a pseudovaluation subring of $S$ , and for any ring $T$ such that $R \subset T \subset S$ , $T$ is a pseudovaluation subring of $S$ . We show that if $S$ is not local, then there no such $T$ exists between $R$ and $S$ . We also characterize maximal non-pseudovaluation subrings of a local integral domain.

Maximal non $λ$ -subrings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with unity. The notion of maximal non $λ$ -subrings is introduced and studied. A ring $R$ is called a maximal non $λ$ -subring of a ring $T$ if $R \subset T$ is not a $λ$ -extension, and for any ring $S$ such that $R \subset S \subseteq T$ , $S \subseteq T$ is a $λ$ -extension. We show that a maximal non $λ$ -subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D + M$ construction is a maximal non $λ$ -domain is given. A necessary condition...

Certain simple maximal subfields in division rings

Mehdi Aaghabali, Mai Hoang Bien (2019)

Czechoslovak Mathematical Journal

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Let $D$ be a division ring finite dimensional over its center $F$ . The goal of this paper is to prove that for any positive integer $n$ there exists $a \in D^{(n)},$ the $n$ th multiplicative derived subgroup such that $F (a)$ is a maximal subfield of $D$ . We also show that a single depth- $n$ iterated additive commutator would generate a maximal subfield of $D .$

Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

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We consider the maximal function $| | (S^{a} f) [x] {| |}_{L^{\infty} [- 1, 1]}$ where $(S^{a} f) {(t)}^{\land} (ξ) = e^{{i t | ξ |}^{a}} f ̂ (ξ)$ and 0 < a < 1. We prove the global estimate $| | S^{a} f {| |}_{L ² (ℝ, L^{\infty} [- 1, 1])} {\leq C | | f | |}_{H^{s} (ℝ)}$ , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.