scholarly journals Pointwise convergence problem of free Benjamin-Ono-Burgers equation

2022 ◽  
Vol 7 (4) ◽  
pp. 5527-5533
Author(s):  
Fei Zuo ◽  
◽  
Junli Shen ◽  
Keyword(s):  
Maximal Function ◽  
Pointwise Convergence ◽  
Burgers Equation ◽  
Convergence Problem ◽  
Function Estimate ◽  
Almost Everywhere

<abstract><p>In this paper, we show the almost everywhere pointwise convergence of free Benjamin-Ono-Burgers equation in $ H^{s}({\bf{R}}) $ with $ s &gt; 0 $ with the aid of the maximal function estimate.</p></abstract>

scholarly journals POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES

2018 ◽  
Vol 6 ◽  
Author(s):  
XIUMIN DU ◽  
LARRY GUTH ◽  
XIAOCHUN LI ◽  
RUIXIANG ZHANG
Keyword(s):  
Lebesgue Measure ◽  
Pointwise Convergence ◽  
General Setting ◽  
Strichartz Estimates ◽  
Convergence Problem ◽  
Fractal Measure ◽  
Almost Everywhere

We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geqslant 3$, $\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$$=f(x)$ almost everywhere with respect to Lebesgue measure for all $f\in H^{s}(\mathbb{R}^{n})$ provided that $s>(n+1)/2(n+2)$. The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞

1977 ◽  
Vol 20 (3) ◽  
pp. 277-284 ◽  
Author(s):  
Richard Duncan
Keyword(s):  
Probability Theory ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Almost Everywhere Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Convergence Results ◽  
Convergence In Norm

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

scholarly journals Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz SpacesΓp,w

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Maciej Ciesielski ◽  
Anna Kamińska
Keyword(s):  
Banach Space ◽  
Weight Function ◽  
Banach Spaces ◽  
Maximal Function ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Lorentz Spaces ◽  
Best Constant ◽  
Decreasing Rearrangement ◽  
Order Continuous

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spacesEand in particular on Lorentz spacesΓp,w={f:∫(f**)pw<∞}for any0<p<∞and a nonnegative locally integrable weight functionw, wheref**is a maximal function of the decreasing rearrangementf*for any measurable functionfon(0,α), with0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages off∈E, whereEis an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximantsfϵoff∈Γp,worf∈Γp-1,w,1<p<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any functionf∈Γp-1,w,1<p<∞.

scholarly journals Pointwise Convergence of the Schrödinger Flow

2020 ◽  
Author(s):  
Erin Compaan ◽  
Renato Lucà ◽  
Gigliola Staffilani
Keyword(s):  
Initial Data ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Homogeneous Part ◽  
Nonlinear Solution ◽  
Nonlinear Schrödinger ◽  
Almost Everywhere ◽  
Smoothing Effects ◽  
Schrodinger Flow

Abstract In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schrödinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing effects for the non-homogeneous part of the solution can be used to upgrade to a uniform convergence to zero of this part, and we discuss the sharpness of the results obtained. We also use randomization techniques to prove that with much less regularity of the initial data, both in continuous and the periodic settings, almost surely one obtains uniform convergence of the nonlinear solution to the initial data, hence showing how more generic results can be obtained.

scholarly journals Pointwise Convergence of Trigonometric Series

1987 ◽  
Vol 43 (3) ◽  
pp. 291-300 ◽  
Author(s):  
Chang-Pao Chen
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Nonnegative Integer ◽  
Pointwise Convergence ◽  
Convergence Problem ◽  
Hardy’S Theorem ◽  
Fatou’S Theorem

We establish two results in the pointwise convergence problem of a trigonometric series for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known L1-convergence classes. A similar result is also established for the case that , where {ln} satisfies certain conditions.

BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures

2010 ◽  
Vol 62 (6) ◽  
pp. 1419-1434
Author(s):  
Dachun Yang ◽  
Dongyong Yang
Keyword(s):  
Growth Condition ◽  
Maximal Function ◽  
Radon Measure ◽  
Maximal Operator ◽  
Maximal Operators ◽  
Open Ball ◽  
Type Space ◽  
Almost Everywhere ◽  
Doubling Measures ◽  
Approximation Of The Identity

AbstractLet μ be a nonnegative Radon measure on ℝd that satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ ℝd and r > 0, μ(B(x, r)) ≤ C0rn, where B(x, r) is the open ball centered at x and having radius r. In this paper, the authors prove that if f belongs to the BMO-type space RBMO(μ) of Tolsa, then the homogeneous maximal function S( f ) (when ℝd is not an initial cube) and the inhomogeneous maximal function ℳS( f ) (when ℝd is an initial cube) associated with a given approximation of the identity S of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, S and ℳS are bounded from RBMO(μ) to the BLO-type space RBLO(μ). The authors also prove that the inhomogeneous maximal operator ℳS is bounded from the local BMO-type space rbmo(μ) to the local BLO-type space rblo(μ).

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