We prove that one can extend any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$BMO^{x}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mi>x</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> function a given in a cube in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^{d+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to become a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$BMO^{x}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mi>x</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\widehat{a}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>a</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^{d+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> almost preserving its <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$[a]^{\sharp }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo stretchy="false">]</mo> </mrow> <mo>♯</mo> </msup> </math></EquationSource> </InlineEquation> seminorm, which is, loosely speaking, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>-norm of the maximal function in t and BMO-norm in x. Bibliography: 7 titles.

EXTENDING BMO FUNCTIONS IN PARABOLIC SETTING

期刊论文
发表日期：

2025年11月1日

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N. V. Krylov

摘要

We prove that one can extend any $BMO^{x}$ $B M O^{x}$ function a given in a cube in $\mathbb {R}^{d+1}$ $R^{d + 1}$ to become a $BMO^{x}$ $B M O^{x}$ functions $\widehat{a}$ $\hat{a}$ in $\mathbb {R}^{d+1}$ $R^{d + 1}$ almost preserving its $[a]^{\sharp }$ ${[a]}^{♯}$ seminorm, which is, loosely speaking, $L_{\infty }$ $L_{\infty}$ -norm of the maximal function in t and BMO-norm in x. Bibliography: 7 titles.