### A direct method for boundary integral equations on a contour with a peak.

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A real-valued Hardy space $H{\xb9}_{\surd}\left(\right)\subseteq L\xb9\left(\right)$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H{\xb9}_{\surd}\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{\xb9}_{\surd}\left(\right)$, and no Orlicz space...

For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral...

Given a function $f\left(t\right)\ge 0$ on $\mathbb{R}$ with ${\int}_{-\infty}^{\infty}\left(f\right(t)/(1+{t}^{2}\left)\right)dt\<\infty $ and $\left|f\right(t)-f({t}^{\text{'}}\left)\right|\le l|t-{t}^{\text{'}}|$, a procedure is exhibited for obtaining on $\u2102$ a (finite) superharmonic majorant of the function$$F\left(z\right):\frac{1}{\pi}{\int}_{-\infty}^{\infty}\frac{\left|\U0001d50dz\right|}{|z-t{|}^{2}}f\left(t\right)dt-Al\left|\U0001d50dz\right|,$$where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $\mathbb{R}$, is such that ${\int}_{-\infty}^{\infty}\left(f\right(t)/(1+{t}^{2})dt\<\infty $ and denote, for any constant $\alpha \>0$ and each $x\in \mathbb{R}$, the unique...

Let $G\subset {\mathbb{R}}^{m}$$(m\ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ be a finite measure on $B$. Consider the space ${L}^{1}\left(\sigma \right)$ of all $\sigma $-integrable functions on $B$ and, for each...

The Fourier problem on planar domains with time moving boundary is considered using integral equations. Solvability of those integral equations in the space of bounded Baire functions as well as the convergence of the corresponding Neumann series are proved.

This paper is concerned with jump conditions for the double layer potential associated with the two-dimensional Helmholtz equation for Hölder continuous boundary data on arbitrary rectifiable Jordan closed curves in ℝ².