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 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 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 A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.

scholarly journals A note on pointwise convergence for the Schrödinger equation

2017 ◽  
Vol 166 (2) ◽  
pp. 209-218 ◽  
Author(s):  
RENATO LUCÀ ◽  
KEITH M. ROGERS
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Schrodinger Equation ◽  
Pointwise Convergence

AbstractWe consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain proved that there is initial data, in Hs(ℝn) with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.

scholarly journals Uniform almost everywhere domination

2006 ◽  
Vol 71 (3) ◽  
pp. 1057-1072 ◽  
Author(s):  
Peter Cholak ◽  
Noam Greenberg ◽  
Joseph S. Miller
Keyword(s):  
Lebesgue Measure ◽  
Almost Everywhere ◽  
A Priority ◽  
Almost All

AbstractWe explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

scholarly journals NORMING SETS AND RELATED REMEZ-TYPE INEQUALITIES

2015 ◽  
Vol 100 (2) ◽  
pp. 163-181 ◽  
Author(s):  
A. BRUDNYI ◽  
Y. YOMDIN
Keyword(s):  
Lebesgue Measure ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
General Setting ◽  
Continuous Functions ◽  
Positive Lebesgue Measure ◽  
Exponential Polynomials ◽  
Absolute Value ◽  
Several Variables

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’,Comm. Inst. Sci. Kharkov13(1936), 9–95] bounds the maximum of the absolute value of a real polynomial$P$of degree$d$on$[-1,1]$through the maximum of its absolute value on any subset$Z\subset [-1,1]$of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of$n$variables’,Math. USSR Izv.37(1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’,Israel. J. Math.186(2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’,J. Approx. Theory162(2010), 72–93]). Still, given a subset$Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here${\mathcal{P}}_{d}(\mathbb{R}^{n})$is the space of real polynomials of degree at most$d$on$\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality:$\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$for all$P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$with$C$independent of$P$. (Although${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree$d$in$\mathbb{R}^{n}$, there are many apparently unrelated reasons for$Z\subset [-1,1]^{n}$to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces$V$of continuous functions on$[-1,1]^{n}$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for$Z$to be$V$-norming, partly known, partly new, restricting ourselves to the simplest nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants$N_{V}(Z)$in the Remez-type inequalities for$V$, as the function of the set$Z$, showing that it is Lipschitz in the Hausdorff metric.

On Continuous Functions which are Partially Differentiable Almost Everywhere

1969 ◽  
Vol 12 (5) ◽  
pp. 668-672
Author(s):  
L.V. Toralballa
Keyword(s):  
Lebesgue Measure ◽  
Surface Area ◽  
Measure Zero ◽  
Continuous Functions ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.

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