Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials

Abstract

Let $0<\sigma <n/2$ and $H=(-\Delta )^\sigma +V(x)$ be Schrödinger type operators on ${\mathbb {R}}^n$ with a class of scaling-critical potentials V(x), which include the Hardy potential $a|x|^{-2\sigma }$ with a sharp coupling constant $a\ge -C_{\sigma ,n}$ ($C_{\sigma ,n}$ is the best constant of Hardy’s inequality of order $\sigma$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schrödinger equation associated with H. In the case of the subcritical coupling constant $a>-C_{\sigma ,n}$, we first prove uniform resolvent estimates of Kato–Yajima type for all $0<\sigma <n/2$, which turn out to be equivalent to Kato smoothing estimates for the Cauchy problem. We then establish Strichartz estimates for $\sigma >1/2$ and uniform Sobolev estimates of Kenig–Ruiz–Sogge type for $\sigma \ge n/(n+1)$. These extend the same properties for the Schrödinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain improved Strichartz estimates with a gain of regularities for general initial data if $1<\sigma <n/2$ and for radially symmetric data if $n/(2n-1)<\sigma \le 1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e., $a=-C_{\sigma ,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.

Appendices

Appendix A. Strichartz Estimates for the Free Evolution

Here we prove Lemmas 3.1 and 3.2. For Lemma 3.1, we in fact prove the following more general result to include the operators $P_0(D)$ considered in Subsection 5.1.

Lemma A.1

Assume either that $H_0=(-\Delta )^\sigma$ with $\sigma >0$ and $\sigma \ne 1/2$ or that $H_0=P_0(D)$ is given by (5.5) with $\sigma \in {\mathbb {N}}$ and $\sigma <n/2$. Then (3.1) and (3.2) are satisfied.

The proof of this lemma relies on the Keel–Tao theorem [34], the Littlewood–Paley square function estimates for $\varphi _j(D)$ given in Subsection 1.2 and the following localized dispersive estimate (with the implicit constant independent of t and j):

$$\begin{aligned} {\Vert e^{-itH_0}\Phi _j(D)\Vert }_{L^1\rightarrow L^\infty }\lesssim 2^{-jn(\sigma -1)}|t|^{-n/2},\quad t\ne 0,\ j\in {\mathbb {Z}}, \end{aligned}$$

(A.1)

where $\Phi _j(\xi )=\Phi (2^{-j}\xi )$ and $\Phi \in C_0^\infty ({\mathbb {R}}^n)$ is supported away from the origin. (A.1) is easy to obtain if $H_0=(-\Delta )^\sigma$. For $H_0$ given by (5.5), an essential ingredient for proving (A.1) is the following decay estimate for the convolution kernel $I(t,x)={\mathcal {F}}^{-1}(e^{-itP_0})(x)$:

$$\begin{aligned} |\partial _x^\alpha I(t,x)| \lesssim {\left\{ \begin{array}{ll} |t|^{-\frac{n+|\alpha |}{2\sigma }}\langle |t|^{-\frac{1}{2\sigma }}x \rangle ^{-\frac{n(\sigma -1)-|\alpha |}{2\sigma -1}} &{}\text {if}\ |t|\lesssim 1,\ \text {or}\ |t|\gtrsim 1\ \text {and}\ |t|^{-1}|x|\gtrsim 1,\\ |t|^{-\frac{n+|\alpha |}{2\delta }}\langle |t|^{-\frac{1}{2\delta }}x \rangle ^{-\frac{n(\delta -1)-|\alpha |}{2\delta -1}} &{}\text {if}\ |t|\gtrsim 1\ \text {and}\ |t|^{-1}|x|\lesssim 1,\end{array}\right. } \end{aligned}$$

(A.2)

where $|\alpha |\le n(\sigma -1)$, $\delta =\min \{j\ |\ a_j>0\}\le \sigma$. This bound follows from [30, Theorem 3.1] by taking $m_2=\sigma ,m_1=\delta$ in this theorem (see also [6] and references therein).

Proof of Lemma A.1

The proof is divided into three steps.

Step 1. We first prove that the following square function estimates for the Littlewood–Paley decomposition $\{\varphi _j(D)\}_{j\in {\mathbb {Z}}}$ hold:

$$\begin{aligned} {\Vert f\Vert }_{L^{q,2}}&\lesssim \Big (\sum _{j\in {\mathbb {Z}}}\Vert \varphi _j(D)f\Vert _{L^{q,2}}^2\Big )^{1/2},\quad 2\le q<\infty , \end{aligned}$$

(A.3)

$$\begin{aligned} {\Vert f\Vert }_{L^{q,2}}&\gtrsim \Big (\sum _{j\in {\mathbb {Z}}}\Vert \varphi _j(D)f\Vert _{L^{q,2}}^2\Big )^{1/2},\quad 1<q\le 2. \end{aligned}$$

(A.4)

These estimates can be easily obtained by the usual estimates and the real interpolation theorem. Indeed, if we set $Sf(j,x):=\varphi _j(D)f(x)$, then the usual square function estimate

$$\begin{aligned} {\Vert Sf\Vert }_{L^2_jL^q_x}\lesssim {\Vert f\Vert }_{L^q_x} \end{aligned}$$

holds for $1<q\le 2$. Then the real interpolation theorem (see Appendix 6.3) implies (A.4). (A.3) follows from (A.4) and a duality argument. Since $e^{-itH_0}$ and $\Gamma _{H_0}$ commute with $\varphi _j(D)$, by virtue of these estimates (A.3) and (A.4), we may assume without loss of generality that $\psi _0=\Phi _j(D)\psi _0,F=\Phi _j(D)F$ with some $\Phi \in C_0^\infty ({\mathbb {R}}^n)$ supported away from the origin so that $\Phi \equiv 1$ on $\mathop {\mathrm {supp}}\nolimits \varphi$.

Step 2. We next prove the dispersive estimate (A.1). Suppose first $H_0=(-\Delta )^\sigma$ and $\sigma \ne 1/2$. Since $|\mathrm {Hess}(|\xi |^{2\sigma })|\sim 1$ on $\mathop {\mathrm {supp}}\nolimits \Phi$, the stationary phase theorem yields

$$\begin{aligned} \Vert e^{-itH_0}\Phi (D)f\Vert _{L^\infty }\lesssim |t|^{-n/2}\Vert f\Vert _{L^1},\quad t\ne 0, \end{aligned}$$

which implies (A.1) by scaling $f(x)\mapsto f(2^jx)$ and the fact that $(-\Delta )^\sigma$ is homogeneous of order $2\sigma$. We next let $\sigma \in {\mathbb {N}}$, $|\alpha |=n(\sigma -1)$ and $H_0$ given by (5.5). When $\delta =\sigma$, we have

$$\begin{aligned} |\partial ^\alpha I(t,x)|\lesssim |t|^{-n/2},\quad t\ne 0, \end{aligned}$$

since $(n+n(\sigma -1))/(2\sigma )=n/2$. When $\delta <\sigma$, we similarly have $|\partial ^\alpha I(t,x)|\lesssim |t|^{-n/2}$ for the former case in (A.2). For the latter case, since $|x|\lesssim |t|$, $|\partial ^\alpha I(t,x)|\lesssim |t|^{-\delta (n,\sigma ,\delta )}$ where

$$\begin{aligned} \delta (n,\sigma ,\delta )=\frac{\sigma n}{2\delta }-\left( 1-\frac{1}{2\delta }\right) \frac{n(\sigma -\delta )}{2\delta -1}=\frac{\sigma n-n\sigma +n\delta }{2\delta }=\frac{n}{2}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} {\Vert |D|^{n(\sigma -1)}e^{-itH_0}\Vert }_{L^1\rightarrow L^\infty }\lesssim |t|^{-n/2},\quad t\ne 0. \end{aligned}$$

(A.5)

which, together with the bound ${\Vert |2^{-j}D|^{n(1-\sigma )}\Phi _j(D)\Vert }_{L^\infty \rightarrow L^\infty }\lesssim 1$, implies (A.1).

Step 3. Now we recall Keel–Tao’s theorem [34, Theorem 10.1] which, in a special case, states that if a family of operators $\{U(t)\}_{t\in {\mathbb {R}}}\subset {\mathbb {B}}(L^2({\mathbb {R}}^n))$ satisfies

• ${\Vert U(t)\Vert }\lesssim 1$ uniformly in $t\in {\mathbb {R}}$;

• ${\Vert U(t)U(s)^*\Vert }_{L^1\rightarrow L^\infty }\lesssim |t-s|^{-\alpha }$ for $t\ne s$ with some $\alpha >0$,

then, for any $\alpha$-admissible pairs (p, q) and $({\tilde{p}},{\tilde{q}})$, one has

$$\begin{aligned} {\Vert U(t)\psi _0\Vert }_{L^p_tL^{q,2}_x}\lesssim {\Vert \psi _0\Vert }_{L^2_x},\quad {\left\| \int _0^tU(t)U(s)^*F(s)ds\right\| }_{L^p_tL^{q,2}_x}\lesssim {\Vert F\Vert }_{L^{{\tilde{p}}'}_tL^{{\tilde{q}}',2}_x}. \end{aligned}$$

By Bernstein’s inequality (C.4) and (A.1), we have

$$\begin{aligned} {\Vert \Phi _j(D)e^{-itH_0}\Phi _j(D)\Vert }_{L^1\rightarrow L^\infty }\lesssim 2^{-jn(\sigma -1)}|t|^{-n/2}. \end{aligned}$$

Putting $N_j=2^{-2j(\sigma -1)}$ and making the change of variable $t\mapsto N_jt$, we have

$$\begin{aligned} {\Vert \Phi _j(D)e^{-iN_j(t-s)H_0}\Phi _j(D)\Vert }_{L^1\rightarrow L^\infty }\lesssim |t-s|^{-n/2}. \end{aligned}$$

By the unitarity of $e^{-itN_jH_0}$ we also obtain ${\Vert \Phi _j(D)e^{-iN_j t H_0}\Vert }\lesssim 1$. Therefore, one can apply the above Keel–Tao theorem to $U(t)=\Phi _j(D)e^{-iN_j t H_0}$ obtaining

$$\begin{aligned} {\Vert e^{-iN_j t H_0}\Phi _j(D)\psi _0\Vert }_{L^{p}_tL^{q,2}_x}&\lesssim {\Vert \psi _0\Vert }_{L^2_x},\\ {\left\| \Phi _j(D)\int _0^te^{-iN_j(t-s)H_0}\Phi _j(D)F(s)ds\right\| }_{L^{p}_tL^{q,2}_x}&\lesssim {\Vert F\Vert }_{L^{{\tilde{p}}'}_tL^{{\tilde{q}}',2}_x}. \end{aligned}$$

By rescaling $t\mapsto N_j^{-1}t$ and $s\mapsto N_j^{-1}s$ and using (C.4), we have (3.1) and (3.2) for $\psi _0,F$ replaced by $\Phi _j(D)\psi _0,\Phi _j(D)F$. Thanks to (A.3) and (A.4), we obtain (3.1) and (3.2). $\square$

Proof of Lemma 3.2

Under the conditions in Lemma 3.2, it has been proved by [23] that

$$\begin{aligned} {\Vert e^{-itH_0}\Phi (D)\psi _0\Vert }_{L^{p_1}_t{\mathcal {L}}^{q_1}_rL^2_\omega }&\lesssim {\Vert \psi _0\Vert }_{L^2_x}, \end{aligned}$$

(A.6)

$$\begin{aligned} {\Vert \Gamma _{H_0}\Phi (D)F\Vert }_{L^{2}_t{\mathcal {L}}^{q_1}_rL^2_\omega }&\lesssim \Vert F\Vert _{L^{2}_t{\mathcal {L}}^{q_2'}_rL^2_\omega }, \end{aligned}$$

(A.7)

where $\Gamma _{H_0}F(t)=\int _0^t e^{-i(t-s)H_0}F(s)ds$. Since $\varphi _j(D)=\Phi _j(D)\varphi _j(D)$, the estimate (A.6) and the same scaling argument as above then imply that

$$\begin{aligned} {\Vert |D|^{s_1}e^{-itH_0}\varphi _j(D)\psi _0\Vert }_{L^{p_1}_t{\mathcal {L}}^{q_1}_rL^2_\omega }\lesssim 2^{-js_1}{\Vert |D|^{s_1}\varphi _j(D)\psi _0\Vert }_{L^2_x} \lesssim {\Vert \varphi _j(D)\psi _0\Vert }_{L^2_x}. \end{aligned}$$

Since $p_1>2$, using Minkowski’s inequality and this estimate, we have

$$\begin{aligned} \Vert |D|^{s_1}e^{-itH_0}\psi _0\Vert _{L^{p_1}_tB[{\mathcal {L}}^{q_1}_rL^2_\omega ]}^2 \lesssim \sum _{j\in {\mathbb {Z}}}\Vert \varphi _j(D)\psi _0\Vert _{L^2_x}^2 \lesssim \Vert \psi _0\Vert _{L^2_x}^2 \end{aligned}$$

and (3.3) follows. Next, since ${\Vert f\Vert }_{{\mathcal {L}}^q_rL^2_\omega }\sim {\Vert f\Vert }_{L^q_x}$ under the radial symmetry, we see that (A.6), (A.7) and the real interpolation theory (see Appendix 6.3) imply

$$\begin{aligned} {\Vert e^{-itH_0}\Phi (D)\psi _0\Vert }_{L^2_tL^{q_1,2}_x}\lesssim {\Vert \psi _0\Vert }_{L^2_x},\quad {\Vert \Gamma _{H_0}\Phi (D)F\Vert }_{L^{2}_tL^{q_1,2}_x}\lesssim {\Vert F\Vert }_{L^{2}_tL^{q_2',2}_x} \end{aligned}$$

for radially symmetric data $\psi _0,F$. The same scaling argument as above then yields

$$\begin{aligned} {\Vert |D|^{s_1}e^{-itH_0}\varphi _j(D)\psi _0\Vert }_{L^2_tL^{q_1,2}_x}&\lesssim {\Vert \varphi _j(D)\psi _0\Vert }_{L^2_x},\\ {\Vert |D|^{s(2,q_1)}\Gamma _{H_0}\varphi _j(D)F\Vert }_{L^{2}_tL^{q_1,2}_x}&\lesssim {\Vert \varphi _j(D)|D|^{-s(2,q_2)}F\Vert }_{L^{2}_tL^{q_2',2}_x}, \end{aligned}$$

which, together with (A.3) and (A.4), implies the desired estimate (3.4). $\square$

Appendix B. Proof of Example 1.1

Let $H_0=P_0(D)$ be given by $P_0(\xi )=\sum _{j=1}^Ja_j|\xi |^{2\sigma _j}$, where $J\in {\mathbb {N}}$, $0<\sigma _1<\sigma _2<\ldots<\sigma _J=\sigma <n/2$, $a_j\ge 0$ and $a_J=1$. Recall that, in such a case, $H_\ell$ are given by

$$\begin{aligned} H_\ell =(2\sigma )^{\ell }(-\Delta )^\sigma +\sum _{j=1}^{J-1}(2\sigma _j)^{\ell }a_j(-\Delta )^{\sigma _j},\quad \ell =0,1,2. \end{aligned}$$

Here we show that the conditions in Example 1.1 implies Assumption A associated with these $H_\ell$. Firstly, (1.17) is just a paraphrase of (1.13). Secondly, we use (1.18) and the condition $a_1,\ldots ,a_{m-1}\ge 0$ to obtain (1.14), namely

$$\begin{aligned} \langle (H_1+V_1)u,u \rangle \ge \langle (2\sigma (-\Delta )^\sigma +V_1)u,u \rangle \gtrsim \langle (-\Delta )^\sigma u,u \rangle . \end{aligned}$$

(B.1)

Finally, writing

$$\begin{aligned} H_2+V_2=2\sigma \Big (2\sigma (-\Delta )^\sigma +\sum _{j=1}^{J-1}\frac{\sigma _j}{\sigma }2\sigma _ja_j(-\Delta )^{\sigma _j} +V_1\Big )-2\sigma V_1+V_2 \end{aligned}$$

and using the fact $\sigma _ja_j\le \sigma a_j$, we have

$$\begin{aligned} |\langle (H_2+V_2)u,u \rangle |\le \langle (2\sigma (H_1+V_1)u,u \rangle +|\langle (2\sigma V_1-V_2)u,u \rangle |. \end{aligned}$$

This bound, together with (1.19) and the first inequality in (B.1), implies (1.15).

Appendix C. Some Supplementary Materials from Harmonic Analysis

Here we record several materials from Harmonic Analysis used frequently in the paper. We refer to textbooks [20] and [1] for details.

• (i) $\underline{\textit{Real interpolation space and theorem}}$. Let $(X_1,X_2)$ be a Banach couple, i.e., $X_1,X_2$ are two Banach spaces continuously embedded into a Hausdorff topological vector space. For $0<\theta <1$ and $1\le q\le \infty$, the real interpolation space $X_{\theta ,q}=(X_1,X_2)_{\theta ,q}$ is a Banach space satisfying $X_0\cap X_1\subset X_{\theta ,q}\subset X_0+X_1$, $X_{\theta ,q}=X_0$ if $X_0=X_1$, $X_{\theta ,q}=X_{1-\theta ,q}$ and

  $$\begin{aligned} X_{\theta ,1}\hookrightarrow X_{\theta ,q_1}\hookrightarrow X_{\theta ,q_2}\hookrightarrow X_{\theta ,\infty },\quad 1<q_1\le q_2<\infty . \end{aligned}$$

  (C.1)

  Let $(X_0,X_1)$ and $(Y_0,Y_1)$ be two Banach couples and T be a bounded linear operator from $(X_0,X_1)$ to $(Y_0,Y_1)$ in the sense that $T:X_j\rightarrow Y_j$ and ${\Vert T\Vert }_{X_j\rightarrow Y_j}\le M_j$ for $j=0,1$. Then T extends to a bounded operator from $X_{\theta ,q}$ to $Y_{\theta ,q}$ satisfying ${\Vert T\Vert }_{X_{\theta ,q}\rightarrow Y_{\theta ,q}}\le M_0^{1-\theta }M_1^\theta$.

• (ii) $\underline{\textit{Lorentz space}}$. Let $(M,d\mu )$ be a $\sigma$-finite measure space. The Lorentz space $L^{p,q}=L^{p,q}(M,d\mu )$ is realized as a real interpolation between Lebesgue spaces, namely $L^{p_\theta ,q}=(L^{p_0},L^{p_1})_{\theta ,q}$ where $1\le p_0<p_1\le \infty$, $1\le q\le \infty$, $1/p_\theta =(1-\theta )/p_0+\theta /p_1$ and $0<\theta <1$. By (C.1), we have the following continuous embeddings:

  $$\begin{aligned} L^{p,1}\hookrightarrow L^{p,q_1}\hookrightarrow L^{p,p}=L^p\hookrightarrow L^{p,q_2}\hookrightarrow L^{p,\infty },\quad 1\le q_1\le p\le q_2\le \infty . \end{aligned}$$

  Moreover, for $1<p,q<\infty$, we have $(L^{p,q})'=L^{p',q'}$ and ${\Vert f\Vert }_{L^{p,q}}\sim \sup _{{\Vert g\Vert }_{L^{p',q'}}=1}\left| \int fgdx\right|$. Finally, the following O’Neil inequality (Hölder’s inequality for Lorentz norms) holds:

  $$\begin{aligned} {\Vert fg\Vert }_{L^{p,q}}\lesssim {\Vert f\Vert }_{L^{p_1,q_1}}{\Vert g\Vert }_{L^{p_2,q_2}},\quad {\Vert fg\Vert }_{L^{p,q}}\lesssim {\Vert f\Vert }_{L^{\infty }}{\Vert g\Vert }_{L^{p,q}} \end{aligned}$$

  (C.2)

  where $1\le p,p_1,p_2<\infty$, $1\le q,q_1,q_2\le \infty$, $1/p_1+1/p_2=1/p$ and $1/q_1+1/q_2=1/q$.

• (iii) $\underline{\textit{Bochner space}}$. Given a Banach space X and $1\le p\le \infty$, the Bochner space $L^pX=L^p(M,d\mu ;X)$ is defined by the norm $\Vert f\Vert _{L^pX}=\Vert \Vert f\Vert _X\Vert _{L^p}$. For any Banach couple $(X_0,X_1)$, $0<\theta <1$, $1<p_0\le p_1<\infty$, the real interpolation space between $L^{p_0}X_0$ and $L^{p_1}X_1$ with the second exponent $q=p_\theta$ is given by $(L^{p_0}X_0,L^{p_1}X_1)_{\theta ,p_\theta }=L^{p_\theta }X_{\theta ,p_\theta }$. In particular, $(L^2_tL^{q_0}_x,L^2_tL^{q_1}_x)_{\theta ,2}=L^2_tL^{q_\theta ,2}_x$ for $1<q_0<q_1<\infty$ and $0<\theta <1$. Note that $(L^{p_0}X_0,L^{p_1}X_1)_{\theta ,q}$ is not necessarily equal to $L^{p_\theta }X_{\theta ,q}$ if $q\ne p_\theta$.

• (iv) $\underline{\textit{Sobolev}'{} \textit{s inequality}}$. If $1<p<q<\infty$, $1<s<n$ and $1/p-1/q=s/n$, then

  $$\begin{aligned} {\Vert f\Vert }_{L^{q,2}({\mathbb {R}}^n)}\lesssim {\Vert |D|^sf\Vert }_{L^{p,2}({\mathbb {R}}^n)}. \end{aligned}$$

  (C.3)

  This inequality follows from the Hardy–Littlewood–Sobolev inequality $|D|^{-s}:L^p\rightarrow L^q$ and the real interpolation theorem.

• (v) $\underline{ {\text {Bernstein's inequality}}}$. Let $\varphi \in C_0^\infty ({\mathbb {R}}^n)$ be supported away from the origin. Then, for all $1\le p\le q\le \infty$, $\varphi _j(D)=\varphi (2^{-j}D)$ satisfies

  $$\begin{aligned} {\Vert \varphi _j(D)\Vert }_{L^{p,2}\rightarrow L^{q,2}}\lesssim 2^{-jn(1/q-1/p)},\quad j\in {\mathbb {Z}}, \end{aligned}$$

  (C.4)

  with $L^{r,2}$ replaced by $L^r$ if $r=1,\infty$. Since $\varphi (D)f={\check{\varphi }}*f$, the special case $j=0$ follows from Young’s convolution inequality and real interpolation theorem. By virtue of the scaling $f(x)\mapsto f(2^jx)$, the general cases also follow from the case $j=0$.

• (vi) $\underline{{\textit{Christ-Kiselev}'{} \textit{s lemma}}}$. Let $-\infty \le a<b\le \infty$, ${\mathcal {X}},{\mathcal {Y}}$ be Banach spaces of functions on ${\mathbb {R}}^n$ so that ${\mathcal {X}}\cap L^2$ is dense in ${\mathcal {X}}$, and $\{K(t,s)\}_{t,s\in (a,b)}\subset {\mathbb {B}}(L^2)$ be such that $K:L^2\rightarrow C((a,b)^2;L^2)$. Define an integral operator T with the operator valued kernel K by

  $$\begin{aligned} TF(t)=\int _a^bK(t,s)F(s)ds. \end{aligned}$$

  Assume $TF(t)\in {\mathcal {Y}}$ for a.e. $t\in (a,b)$ and there exist $1\le p<q\le \infty$ and $C>0$ such that

  $$\begin{aligned} \Vert TF\Vert _{L^q((a,b);{\mathcal {Y}})}\le C\Vert F\Vert _{L^p((a,b);{\mathcal {X}})} \end{aligned}$$

  (C.5)

  for any simple function $F:(a,b)\rightarrow L^2\cap {\mathcal {X}}$. Then the operator

  $$\begin{aligned} {\tilde{T}}f(t)=\int _a^tK(t,s)F(s)ds \end{aligned}$$

  satisfies the following estimate for the same p, q:

  $$\begin{aligned} \Vert {\tilde{T}}F\Vert _{L^q((a,b);{\mathcal {Y}})}\le {\tilde{C}}\Vert F\Vert _{L^p((a,b);{\mathcal {X}})}, \end{aligned}$$

  where ${\tilde{C}}=C2^{1-2(1/p-1/q)}(1-2^{-(1/p-1/q)})^{-1}$. Note that the condition $p<q$ is necessary since ${\tilde{C}}\rightarrow \infty$ as $p\rightarrow q$. This is a minor modification of [52, Lemma 3.1] (see also the original paper [7]) where the condition $K\in C({\mathbb {R}}^2;{\mathbb {B}}({\mathcal {X}},{\mathcal {Y}}))$ was assumed to define $T,{\tilde{T}}$ on $C_t{\mathcal {X}}\cap L^1_t{\mathcal {X}}$. In the present setting, the above assumption is sufficient to define $T,{\tilde{T}}$ on $C_t({\mathcal {X}}\cap L^2)\cap L^1_t({\mathcal {X}}\cap L^2)$ and the same proof as that of [52, Lemma 3.1] works well to obtain the above statement. Such a modification is useful when one considers the case with $K(t,s)=e^{-i(t-s)H}$ to prove inhomogeneous Strichartz estimates for ${\tilde{T}}=\Gamma _H$ by using the corresponding homogeneous Strichartz estimates for $e^{-itH}$, since $e^{-i(t-s)H}:L^2_x\rightarrow C({\mathbb {R}}^2;L^2_x)$ for any self-adjoint operator H on $L^2$, while it is not always true that $e^{-itH}: {\mathcal {X}}\rightarrow {\mathcal {Y}}$ for each t unless ${\mathcal {X}}={\mathcal {Y}}=L^2$. Moreover, the condition that $TF(t)\in {\mathcal {Y}}$ for a.e. t follows from the corresponding homogeneous Strichartz estimates for $e^{-itH}$.