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The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

For each $p\ge 1$ with $f_0\in L^p$, a sufficient condition is $h\in L^1([0,T];L^p({\mathbb{R}}^2))$ on every finite interval. The Minkowski integral inequality and measure preservation give the finite-time Lp bound for Hamiltonian transportFor all finite $p$ simultaneously, impose, for example, $h\in L_{\mathrm{loc}}^1([0,\infty );L^1\cap L^{\infty })$. The elementary bound $\Vert h_s{\Vert }_p\le \Vert h_s{\Vert }_1^{1/p}\Vert h_s{\Vert }_{\infty }^{1-1/p}$ and the Holder inequality in time show that its $L^p$ norms are locally integrable for every $1\le p<\infty$. If a bounded initial value is also required, the same assumption controls the $p=\infty$ endpoint by the previous part.

A concrete stronger condition, compatible with a nonzero $C^1$ source, is that on each finite time interval the source has a common compact support in $(x,v)$. Its continuity then makes it bounded on that compact cylinder, so all these integrability conditions hold. A nonzero smooth source compactly supported in phase space supplies examples.

Use the genuine planar Givens rotationIn the second component the cosine multiplies $v_j$: the repeated $v_i$ in the printed formula is an error. With that printed expression, at $\theta =0$ the pair becomes $(v_i,v_i)$, which does not preserve length or measure. The rotation-based claims require the corrected expression. Also take $N\ge 2$, since the normalization by $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ is undefined for $N=1$.

Let $C_N=\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ and $U_{ij,\theta }F=F\circ R_{ij,\theta }$. The change of variables formula and determinant one give $\Vert U_{ij,\theta }F{\Vert }_2=\Vert F{\Vert }_2$. Thus each is a unitary operator, with adjoint $U_{ij,-\theta }$. The Kac collision operator is the averageThe Minkowski integral inequality gives $\Vert QF{\Vert }_2\le \Vert F{\Vert }_2$, so $Q$ is bounded. For the Hilbert space inner product, integration and the angular change $\theta \mapsto -\theta$ giveHence $Q=Q^{\ast }$ and $\Vert Q\Vert \le 1$. In fact a nonzero radial Gaussian function is fixed by every rotation, showing $\Vert Q\Vert =1$. Angular averages can be understood as strong Bochner integrals; continuity of rotations in $L^2$ follows first for smooth compactly supported functions, then by density.

For the globally invertible Hamiltonian flow from (a), the general characteristic formula isEvery $\Phi (s,t)$ preserves Lebesgue measure, by (b). Thus composition with it is an isometry of each Lp space. The Minkowski integral inequality givesApplying the Minkowski inequality also in time yields the finite-time Lp bound for Hamiltonian transport:The smoothness assumptions allow the characteristic construction; the norm estimate itself only uses the Lp space data and volume preservation.

For a concrete failure on infinite time, choose $\omega =1$, $f_0=0$, andThis is smooth, time independent and in every finite Lp space; it is invariant under the isotropic harmonic oscillator flow. Therefore the solution grows linearly:This invariant-source secular growth in Hamiltonian transport supplies the counterexample even with zero initial data.

Denote the damped free-transport evolution from time $s$ to time $t$ byBy the argument in (b), $\Vert U_a(t,s)g{\Vert }_2\le \Vert g{\Vert }_2$. The integral operator is the Boltzmann Volterra operatorThe Minkowski integral inequality and (a) give the useful stronger pointwise boundTherefore the requested estimate isFor strongly measurable bounded $L^2$-valued $f$, these are Bochner integrals; their finite norm bounds establish existence of the integrals.

The Sobolev fundamental theorem of calculus on lines gives, for almost every $x$,Apply the Minkowski integral inequality and translation invariance of Lebesgue measure:The restriction on $h$ ensures that every translated copy used above lies in $U$.

A kernel for density estimation is an integrable function $K$ with $\int K=1$. With $K_h(u)=h^{-1}K(u/h)$, the kernel density estimator isA kernel of order ell satisfies

Split the integrated $L^p$ risk into stochastic and bias terms:For fixed $x$, put $A_i(x)=K_h(x-X_i)$ and $W_i(x)=A_i(x)-\mathbb{E}{A}_i(x)$. The supplied Rosenthal inequality givesNowAlso $\mathbb{E}{W}_1(x{)}^2\le (K_h^2\ast f)(x)$. The Young convolution inequality with exponent $p/2$ givesAfter integration and multiplication by the outer factor $2^{p-1}$, the stochastic contribution is at most

It remains to control the bias. Here $\ell =\lceil \beta \rceil =m+1$. The Taylor formula with integral remainder, the vanishing kernel moments, and Minkowski integral inequality giveThe defining Nikol'skii smoothness bound is $L|hu{|}^{\beta -m}$, soRaising to the $p$th power and applying the outer factor $2^{p-1}$ proves

Tonelli's theorem and the triangle inequality giveApplying Holder inequality in the $y$ variable for each fixed $x$ yieldsand $\Vert g{\Vert }_q\le 1$ gives the claimed inequality. This is the duality proof of the Minkowski integral inequality.

For $f_z(y)=f(yz)$, the substitution $u=yz$ givesHomogeneity of $K$ and the substitution $x=yz$ giveApply part (a), or equivalently the Minkowski integral inequality, to this integral and use the preceding scaling identity:

Smooth forcing need not be integrable in phase space. For the hyperbolic characteristic flow for an inverted oscillator, the source $x^2+v^2$ accumulates as $q_t=\frac{1}{2}\sinh (2t)(x^2+v^2)-(\cosh (2t)-1)xv$. Its smallest quadratic-form eigenvalue is $(1-e^{-2t})/2>0$ for $t>0$. Starting with a nonnegative integrable Gaussian function, the solution is therefore unbounded and outside every finite Lp space at positive times. Locally time-integrable $L^p$ forcing prevents this failure by the Minkowski integral inequality.