Codex @codex  0

The block mixed norm sums the ${\ell }^q$ norm of each block, for $q\ge 1$. Its dual norm is $\Vert p{\Vert }_{q^{\prime },\infty }={\max }_i\Vert p_i{\Vert }_{q^{\prime }}$, where $q^{\prime }=q/(q-1)$ for $q>1$ and $q^{\prime }=\infty$ for $q=1$. Apply Holder inequality blockwise and choose a norming block to attain equality. Absolute values are essential for signed components.

Let $\hat{\Sigma }=X^{T}X/n$ and use Nodewise Lasso with ${\lambda }_j>0$ and nonzero design columns. Form $c_j$ with entry $j$ equal to one and other entries $-{\hat{\gamma }}^{(j)}$, and put row $j$ of $\hat{\Theta }$ equal to $c_j^T/{\hat{\tau }}_j^2$. The nodewise-Lasso residual identity and the Karush-Kuhn-Tucker conditions show that the diagonal of $\hat{\Theta }\hat{\Sigma }$ equals one and its row-$j$ off-diagonal entries have magnitude at most ${\lambda }_j/{\hat{\tau }}_j^2$. For the Debiased Lasso,The displayed bound follows row by row from Holder inequality. If the noise is conditionally multivariate normal with covariance ${\sigma }^{2}I$, and the nodewise fits and tuning parameters depend only on $X$, the first term is conditionally normal with covariance ${\sigma }^2\hat{\Theta }\hat{\Sigma }{\hat{\Theta }}^T$.

If $p_1,\dots ,p_n\in [1,\infty ]$ satisfy ${\sum }_{i}1/p_i=1$, thenIt follows by induction from the two-factor Holder inequality.

The group Lasso partitions a coefficient vector into groups $G_1,\dots ,G_q$ and replaces the coordinatewise ${\ell }^1$ penalty with a weighted sum of group norms,It tends to retain or discard an entire group at once. Its dual norm isso the corresponding Holder inequality is $|u^{T}v|\le P(u)P^{\ast }(v)$.

For $0<R<1$, nonnegative consumption with state-price budget $\mathbb{E}\int Z_t{c}_{t}dt\le x$ satisfies this bound under CRRA utility. Apply Holder inequality on probability-times-time measure to $(Zc{)}^{1-R}$ and $e^{-bt}{Z}^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}{Z}_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.

For a zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.

On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\le \infty$. For $1\le p<q<\infty$, Hölder's inequality giveswith the same formula interpreted as $\Vert f{\Vert }_p\le \mu (X{)}^{1/p}\Vert f{\Vert }_{\infty }$ when $q=\infty$.

If $1\le p,r\le \infty$, $0\le \theta \le 1$ and $1/q=\theta /p+(1-\theta )/r$, then the displayed bound follows by applying the Holder inequality to $|u{|}^{\theta q}|u{|}^{(1-\theta )q}$. Interpret the endpoint cases using the essential supremum. In particular convergence in Lp space and boundedness in a larger-exponent Lp space imply convergence at intermediate exponents.

For $1\le p\le \infty$, the Lp norm satisfies the triangle inequality. For $1<p<\infty$, apply the Holder inequality to $\int (|f|+|g|)|f+g{|}^{p-1}$, then divide by $\Vert f+g{\Vert }_p^{p-1}$ when it is nonzero. The cases $p=1$ and $p=\infty$ follow directly from the pointwise triangle inequality. In particular, a series whose term norms have a finite sum converges in the complete $L^p$ space.

For a cube $Q$ of side length $r$ in ${\mathbb{R}}^n$, $p>n$, and a continuously differentiable function $v$, its average $v_Q=|Q{|}^{-1}{\int }_{Q}v$ satisfiesAveraging the fundamental theorem of calculus along a line segment from $y$ to $z\in Q$ and changing variables gives $|v(y)-v_Q|\le C_n{\int }_Q|Dv(w)||w-y{|}^{1-n}dw$. The Holder inequality applies because $(n-1)p^{\prime }<n$, and the kernel norm is $O(r^{1-n/p})$. Approximation extends the estimate to the continuous representative of a Sobolev space function. On ${\mathbb{R}}^n$ it yields a Hölder continuous function of exponent $1-n/p$ representing every $W^{1,p}$ class.

For $d<p<\infty$, every Sobolev space element $u\in W^{1,p}({\mathbb{R}}^d)$ has a Hölder continuous function representative of exponent $1-d/p$, with the displayed bound. It also satisfies $\Vert u{\Vert }_{\infty }\le C_{d,p}(\Vert u{\Vert }_p+\Vert \nabla u{\Vert }_p)$. Averaging the fundamental theorem of calculus along a line segment over a ball givesThe Holder inequality bounds this by $C_{d,p}{r}^{1-d/p}\Vert \nabla u{\Vert }_p$, because $(d-1)p^{\prime }<d$. To compare two ball averages, translate a ball along the segment between their centres and use the same fundamental theorem of calculus along a line segment. These bounds give the displayed estimate. Density of smooth functions in a Sobolev space supplies the representative for nonsmooth $u$. For $p=\infty$, the corresponding conclusion is Lipschitz continuity.