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For a uniformly elliptic divergence-form elliptic operator with lower-order coefficients in the supercritical Lebesgue spaces, truncation tests and Moser iteration bound the positive part of a weak subsolution on an interior ball by its Lp norm on a larger ball and the forcing norms. In particular, every weak subsolution of the homogeneous equation is locally bounded above. This is the preliminary estimate needed when a strong maximum principle is phrased using essential suprema.

For $1\le p<\infty$, the space ${\ell }^p$ consists of scalar sequences $x=(x_n)$ satisfyingIt is a Banach space, and ${\ell }^q\subset {\ell }^p$ when $1\le q<p$.

The mass-critical focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$ isIts scaling $u(t,x)\mapsto {\lambda }^{d/2}u({\lambda }^{2}t,\lambda x)$ preserves the Lebesgue space norm $\Vert u{\Vert }_{L^2}$, which is the conserved mass.

A weak solution is an equivalence class $u\in L^{\infty }((0,\infty )\times \mathbb{R})$ satisfying, for every test function $\phi \in C_c^{\infty }([0,\infty )\times \mathbb{R})$,This is integration by parts for the nonconservative transport equation. In particular, the weak formulation contains $c_x\phi$ as well as $c{\phi }_x$. The last integral encodes the initial condition; a pointwise boundary value of an arbitrary Lebesgue space representative is not the definition. Compact support of the test function makes the identity meaningful even if $c$ is unbounded.

Work on a filtered probability space with the usual conditions for a filtration. A continuous martingale belongs to the class of L2-bounded continuous martingales whenThis means a uniform bound over the entire time interval, not merely being square-integrable at each individual time. The L2 martingale convergence theorem gives a terminal variable $M_{\infty }$ and almost sure convergence and convergence in the Lebesgue space $L^2$; moreover $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$ by conditional expectation.

The predictable sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is the smallest sigma-algebra making every left-continuous adapted process measurable. Equivalently, it is generated bywhere the sample coordinate comes first. A previsible process is precisely a process measurable for this predictable sigma-algebra. Values at time zero matter for its definition, although they make no contribution to integration against a continuous quadratic variation starting at zero.

Write $N_2=\Vert M_{\infty }{\Vert }_2$ and $N_{\ast }=\Vert {\sup }_{t\ge 0}|M_t|{\Vert }_2$. The L2 martingale convergence theorem ensures the existence of $M_{\infty }$ and yields $|M_{\infty }|\le {\sup }_t|M_t|$ almost surely. Hence $N_2\le N_{\ast }$.

For a finite horizon $T$, the Doob L2 maximal inequality for the submartingale $|M|$ givesThe inequality does not require $M_0=0$. Send $T$ to infinity: the left side increases to $N_{\ast }^2$ by the monotone convergence theorem, and the right side tends to $4N_2^2$ by convergence in the Lebesgue space $L^2$. Thus the equivalent terminal and maximal norms for L2-bounded continuous martingales satisfyBoth are genuine norms on L2-bounded continuous martingales modulo indistinguishability of stochastic processes: a zero terminal norm gives $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]=0$, first at rational times and then at all times by continuity. This proves that the two norms are equivalent norms.

For $1\le p<\infty$, the Lebesgue space $L^p(X)$ consists of equivalence classes modulo equality almost everywhere of measurable functions satisfyingThe space $L^{\infty }(X)$ consists of essentially bounded measurable functions modulo equality almost everywhere, with norm

Let $1\le p<q<\infty$ and put $r=q/p>1$. By Hölder's inequality,Taking $p$th roots givesFor $q=\infty$, directly integrating $|f{|}^p\le \Vert f{\Vert }_{\infty }^p$ gives the same endpoint estimate. Hence $L^q(X)\subseteq L^p(X)$ when $\mu (X)<\infty$, as in Lp inclusion on a finite measure space.

No. In the Lebesgue space $L^1([0,2\pi ])$, letFor every $g\in L^{\infty }([0,2\pi ])$, finiteness of the interval gives $g\in L^1([0,2\pi ])$, and the Riemann-Lebesgue lemma yieldsThe duality of Lp spaces therefore shows that $f_n\rightharpoonup 0$ in $L^1$. On the other hand,for every $n$. Thus weak convergence does not imply norm convergence in this $L^1$ space; the Weakly null sine sequence in L1 is the required counterexample.

For $1\le p<\infty$, choose nonzero $\varphi \in C_c({\mathbb{R}}^{n-1})$ and $\psi \in C_c(\mathbb{R})$ with $\psi (0)=1$. DefineThese functions are continuous and belong to the Lebesgue space $L^p({\mathbb{R}}^n)$, while their restrictions to the hyperplane are all the same:Changing variables in the normal coordinate givesIf the claimed bounded $T$ existed, thena contradiction. This scaling argument is the failure of an Lp hyperplane trace.

If $f=g$ almost everywhere, then $|f|=|g|$ almost everywhere, so the integral defining $\Vert [f]{\Vert }_1$ is independent of the representative. Homogeneity and the triangle inequality follow from the corresponding pointwise properties of absolute value. Moreover,so this is a norm on the Lebesgue space of almost-everywhere equivalence classes.

The Riesz-Fischer theorem states that $L^p$ is complete for $1\le p\le \infty$. Applied with $p=1$, it says that every $\Vert \cdot {\Vert }_1$-Cauchy sequence of these equivalence classes converges in ${\mathcal{L}}^1({\mathbb{R}}^n)$. Thus ${\mathcal{L}}^1({\mathbb{R}}^n)$ is a Banach space.

For a L2-bounded continuous martingale $M$, define a finite measure on the predictable sigma-algebra by ${\nu }_M(A)=\mathbb{E}{\int }_0^{\infty }{1}_{A}d[M]$. Its total mass is $\mathbb{E}[M{]}_{\infty }=\mathbb{E}{M}_{\infty }^2-\mathbb{E}{M}_0^2$. The Lebesgue space $L^2({\nu }_M)$ identifies predictable processes that agree ${\nu }_M$-almost everywhere. Integration against $M$ is an isometry from this space into L2-bounded continuous martingales starting at zero by the Itô isometry. This gives the correct integrand space even when the quadratic variation is random or is not absolutely continuous in time.