A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Adam Osękowski

Displaying similar documents to “A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions”

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

Similarity:

We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

Similarity:

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

Similarity:

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

On weak sharp minima for a special class of nonsmooth functions

Marcin Studniarski (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by $f (x) : = m a x f_{i} (x) | i = 1, . . ., p$ , where the functions $f_{i}$ are strictly differentiable. It is given in terms of the gradients of $f_{i}$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let ${(h_{k})}_{k \geq 0}$ be the Haar system on [0,1]. We show that for any vectors $a_{k}$ from a separable Hilbert space and any $ε_{k} \in [- 1, 1]$ , k = 0,1,2,..., we have the sharp inequality $| | \sum_{k = 0}^{n} ε_{k} a_{k} h_{k} {| |}_{W ([0, 1])} \leq 2 | | \sum_{k = 0}^{n} a_{k} h_{k} {| |}_{L^{\infty} ([0, 1])}$ , n = 0,1,2,..., where W([0,1]) is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${| | Y | |}_{W (Ω)} {\leq 2 | | X | |}_{L^{\infty} (Ω)}$ , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

Gradient estimates for the p(x)-Laplacian equation in $ℝ^{N}$

Chao Zhang, Shulin Zhou, Bin Ge (2015)

Annales Polonici Mathematici

Similarity:

Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in $ℝ^{N}$ .

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

$L^{q}$ -approach to weak solutions of the Oseen flow around a rotating body

Stanislav Kračmar, Šárka Nečasová, Patrick Penel (2008)

Banach Center Publications

Similarity:

We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in $L^{q}$ -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions...

Extensions of weak type multipliers

P. Mohanty, S. Madan (2003)

Studia Mathematica

Similarity:

We prove that if $Λ \in M_{p} (ℝ^{N})$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_{p} (ℝ^{N})$ is the space of multipliers of $L^{p} (ℝ^{N})$ .

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

Similarity:

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...

Interior $C^{1,\alpha}$ -Regularity of Weak Solutions to the Equations of Stationary Motions of Certain Non-Newtonian Fluids in Two Dimensions

Jorg Wolf (2007)

Bollettino dell'Unione Matematica Italiana

Similarity:

In the present work we prove the interior Hölder continuity of the gradient matrix of any weak solution of equations, which describes the motion of non-Newtonian fluid in two dimensions, restricting ourself to the shear thinning case $1<q<2$ .

The structure of Eberlein, uniformly Eberlein and Talagrand compact spaces in Σ( $R^{r}$ )

V. Farmaki (1987)

Fundamenta Mathematicae

Similarity:

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

Yoichi Uetake (2001)

Studia Mathematica

Similarity:

Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨ \cdot, \cdot ⟩_{X}$ . For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨ {(I - z A)}^{- 1} b, c ⟩_{X}$ . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

Similarity:

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 $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

Similarity:

We give some rather weak sufficient condition for $L^{p}$ boundedness of the Marcinkiewicz integral operator $μ_{Ω}$ on the product spaces $ℝ ⁿ \times ℝ^{m}$ (1 < p < ∞), which improves and extends some known results.

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

Similarity:

We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

Weak lineal convexity

Christer O. Kiselman (2015)

Banach Center Publications

Similarity:

A bounded open set with boundary of class C¹ which is locally weakly lineally convex is weakly lineally convex, but, as shown by Yuriĭ Zelinskiĭ, this is not true for unbounded domains. The purpose here is to construct explicit examples, Hartogs domains, showing this. Their boundary can have regularity $C^{1, 1}$ or $C^{\infty}$ . Obstructions to constructing smoothly bounded domains with certain homogeneity properties will be discussed.

On bases, compactness and weak convergence in the Banach space $A_{p}$

J. T. Marti (1971)

Colloquium Mathematicae

Similarity:

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz, Konrad Zdanowski (2011)

Fundamenta Mathematicae

Similarity:

We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

Similarity:

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

On weakly $θ$ -continuous functions

F. Commaroto, T. Noiri (1986)

Matematički Vesnik

Similarity:

The power set of ω Elementary submodels and weakenings of CH

István Juhász, Kenneth Kunen (2001)

Fundamenta Mathematicae

Similarity:

We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C ₂^{s} (ω ₂)$ , but does not follow from $C ₂^{s} (ω ₂)$ , or even $C^{s} (ω ₂)$ .

Weak and strong density results for the Dirichlet energy

Mariano Giaquinta, Domenico Mucci (2004)

Journal of the European Mathematical Society

Similarity:

Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^{n} \times 𝒴$ with equibounded Dirichlet energies, $B^{n}$ being the unit ball in $ℝ^{n}$ . More precisely, weak limits of graphs of smooth maps $u_{k} : B^{n} \to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2, 1} (B^{n} \times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2, 1} (B^{n} \times 𝒴)$ is the...