\[ \DeclareMathOperator{\ess}{ess} \]

Let \(p\in (0,\infty]\) and \((\Omega, \Sigma, \mu)\) a meausurable space . A function \(f\) is in \(L^p(\mu)\) if

\[ \lVert f\rVert_{L^p(\Omega,\mu)}=\Bigl(\int_{\Omega} \lvert f\rvert^p d\mu\Bigr)^{1/p}. \]

is finite. For \(p=\infty\) we set

\[ \lVert f\rVert_{L^\infty (\Omega,\mu)}= \ess \sup \lvert f\rvert . \]

Special cases Link to heading

• finite dimensions

Inequalities and Embeddings Link to heading

General case Link to heading

• Hölders inequality
• If \(\mu(\Omega)<\infty\) then \(L^p \hookrightarrow L^q\) if \(p>q\).

Euclidean case Link to heading

Whole \(\mathbb{R}^n\) Link to heading

• \(\mathcal{S}(\mathbb{R}^n)\subseteq L^p(\mathbb{R}^n)\)
• Youngs inequality

Interval \(I\subseteq \mathbb{R}\) Link to heading

• For special functions we have \(E \subset I \implies \lVert \cdot\rVert_{L^p(I)} \lesssim \lVert \cdot\rVert_{L^p(E)}\)

See also Link to heading

• locally integrable function
• \(L^p(l^q)\) space